2024年10月20日日曜日

241020

PARI


A377145等

出力してみた。

N=10;

a(n, k) = sum(j=0, n, binomial(j+k, k)*binomial(j, n-j)^2);
b(n) = sum(k=0, N, a(k, n)*x^k + x*O(x^N));

a089627(n, k) = n!/((n-2*k)!*k!^2);
c(n) = sum(k=0, n\2, a089627(n, k)*(1-x-x^2)^(n-2*k)*x^(3*k))/((1-x-x^2)^2-4*x^3 + x*O(x^N))^(n+1/2);
for(n=0, 10, if(b(n)==c(n), print(n, " ", Vec(b(n)))));

出力結果
0 [1, 1, 2, 5, 11, 26, 63, 153, 376, 931, 2317]
1 [1, 2, 5, 16, 44, 122, 341, 940, 2581, 7064, 19258]
2 [1, 3, 9, 34, 111, 351, 1103, 3384, 10224, 30536, 90222]
3 [1, 4, 14, 60, 225, 796, 2764, 9304, 30580, 98700, 313422]
4 [1, 5, 20, 95, 400, 1561, 5915, 21610, 76585, 265075, 898622]
5 [1, 6, 27, 140, 651, 2772, 11354, 44640, 169371, 624742, 2248575]
6 [1, 7, 35, 196, 994, 4578, 20118, 84540, 341397, 1335103, 5078227]
7 [1, 8, 44, 264, 1446, 7152, 33516, 149688, 640233, 2642992, 10582220]
8 [1, 9, 54, 345, 2025, 10692, 53163, 251163, 1133055, 4919200, 20662785]
9 [1, 10, 65, 440, 2750, 15422, 81015, 403260, 1911910, 8701550, 38231765]
10 [1, 11, 77, 550, 3641, 21593, 119405, 624052, 3099811, 14747733, 67602249]
投稿者 時刻: 0 件のコメント:
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2024年9月18日水曜日

240918

PARI


(1/x) * Series_Reversion( x * f(x) ).

a(n) = (1/(n+1)) * [x^n] (1/f(x))^(n+1) となることを確認する。

N=20;
x='x+O('x^(N+2));

f = 1/(1 + 5*x + x^2);
a = sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k);
b = serreverse(x*f)/x;
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

f = 1 + 5*x + x^2;
a = sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k);
b = serreverse(x*f)/x;
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

f = 1/(3 - 2*exp(x));
a = serlaplace(sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k));
b = serlaplace(serreverse(x*f)/x);
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

f = 3 - 2*exp(x);
a = serlaplace(sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k));
b = serlaplace(serreverse(x*f)/x);
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

出力結果
[1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, 5606255, 34578292, 215322310, 1351978807, 8550394455, 54419811354, 348309105300, 2240486766555, 14476490777175, 93914850905862, 611489638708140, 3994697746533171, 0]
[1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, 5606255, 34578292, 215322310, 1351978807, 8550394455, 54419811354, 348309105300, 2240486766555, 14476490777175, 93914850905862, 611489638708140, 3994697746533171]
[]
[1, -5, 49, -600, 8228, -120890, 1860738, -29616675, 483483055, -8050547505, 136201191477, -2334603084060, 40456777529028, -707614530573800, 12475631038479848, -221479557759651840, 3955846194794651388, -71035215358997870750, 1281683885259534013950, -23224445624372453734350, 422457775863392772285090, 0]
[1, -5, 49, -600, 8228, -120890, 1860738, -29616675, 483483055, -8050547505, 136201191477, -2334603084060, 40456777529028, -707614530573800, 12475631038479848, -221479557759651840, 3955846194794651388, -71035215358997870750, 1281683885259534013950, -23224445624372453734350, 422457775863392772285090]
[]
[1, -2, 6, -14, -82, 2058, -22778, 55186, 4737630, -141417830, 1940770358, 18532189410, -2031211536242, 63333969461098, -624072759994266, -46503997775007182, 3129486110236404926, -90436659990999596742, -559947921342589721450, 214981434079884273561730, -12114419920158647173081362, 0]
[1, -2, 6, -14, -82, 2058, -22778, 55186, 4737630, -141417830, 1940770358, 18532189410, -2031211536242, 63333969461098, -624072759994266, -46503997775007182, 3129486110236404926, -90436659990999596742, -559947921342589721450, 214981434079884273561730, -12114419920158647173081362]
[]
[1, 2, 18, 302, 7562, 253542, 10685794, 543309230, 32378850042, 2214215333750, 170939286647570, 14707184259036414, 1395561779648175274, 144795755972202587462, 16308198003201872476866, 1981633767850818093910094, 258406311809937562215099482, 35994776359231593721760238102, 5334228760604574619939352424562, 837995152675419667859461838034590, 139110781237167610980642890783413770, 0]
[1, 2, 18, 302, 7562, 253542, 10685794, 543309230, 32378850042, 2214215333750, 170939286647570, 14707184259036414, 1395561779648175274, 144795755972202587462, 16308198003201872476866, 1981633767850818093910094, 258406311809937562215099482, 35994776359231593721760238102, 5334228760604574619939352424562, 837995152675419667859461838034590, 139110781237167610980642890783413770]
[]
投稿者 時刻: 0 件のコメント:
ラベル:

2024年8月7日水曜日

240807

PARI


A144088

出力してみた。
ついでにRow sums も計算した。

T(n, k) = n!/k!*sum(j=0, n-k, binomial(j, n-k-j)/(n-k-j)!);
for(n=0, 10, for(k=0, n, print1(T(n, k),", ")); print)

print

for(n=0, 30, print1(sum(k=0, n, T(n, k)),", "))

出力結果
1, 
1, 1, 
4, 2, 1, 
18, 12, 3, 1, 
108, 72, 24, 4, 1, 
780, 540, 180, 40, 5, 1, 
6600, 4680, 1620, 360, 60, 6, 1, 
63840, 46200, 16380, 3780, 630, 84, 7, 1, 
693840, 510720, 184800, 43680, 7560, 1008, 112, 8, 1, 
8361360, 6244560, 2298240, 554400, 98280, 13608, 1512, 144, 9, 1, 
110557440, 83613600, 31222800, 7660800, 1386000, 196560, 22680, 2160, 180, 10, 1, 

1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231, 3405357682, 53334454417, 896324308634, 16083557845279, 306827170866106, 6199668952527617, 132240988644215842, 2968971263911288999, 69974827707903049154, 1727194482044146637521, 44552237162692939114282, 1198605668577020653881647, 33572577967799567547243274, 977421343777135316364285889, 29533262279404214911002168626, 924841298668309602644434087927, 29976944827211469262961974756882, 1004499603594644578398078136286609, 34759052263955593644926343695227834, 1240758969214239528262796909096631871, 
投稿者 時刻: 0 件のコメント:
ラベル:

2024年7月20日土曜日

240720

Ruby


A292105

正n 角形において、ちょうどk 本の対角線が交わる点の数を出力してみた。

def d(n, k)
  n % k == 0 ? 1 : 0
end

def a2(n)
  s  = (n ** 3 - 6 * n * n + 11 * n - 6) / 24r
  s += (- 5 * n * n + 46 * n - 72) / 16r * d(n, 2)
  s -= 9 / 4r * d(n, 4)
  s += (-19 * n + 110) / 2r * d(n, 6)
  s +=  54 * d(n, 12)
  s +=  84 * d(n, 18)
  s +=  50 * d(n, 24)
  s -=  24 * d(n, 30)
  s -= 100 * d(n, 42)
  s -= 432 * d(n, 60)
  s -= 204 * d(n, 84)
  s -= 144 * d(n, 90)
  s -= 204 * d(n, 120)
  s -= 144 * d(n, 210)
  (n * s).to_i
end

def a3(n)
  s  = (5 * n * n - 48 * n + 76) / 48r * d(n, 2)
  s += 3 / 4r * d(n, 4)
  s += (7 * n - 38) / 6r * d(n, 6)
  s -=   8 * d(n, 12)
  s -=  20 * d(n, 18)
  s -=  16 * d(n, 24)
  s -=  19 * d(n, 30)
  s +=   8 * d(n, 42)
  s +=  68 * d(n, 60)
  s +=  60 * d(n, 84)
  s +=  48 * d(n, 90)
  s +=  60 * d(n, 120)
  s +=  48 * d(n, 210)
  (n * s).to_i
end

def a4(n)
  s  = (7 * n - 42) / 12r * d(n, 6)
  s -= 5 / 2r * d(n, 12)
  s -=  4 * d(n, 18)
  s +=  3 * d(n, 24)
  s +=  6 * d(n, 42)
  s += 34 * d(n, 60)
  s -=  6 * d(n, 84)
  s -=  6 * d(n, 120)
  (n * s).to_i
end

def a5(n)
  s  = (n - 6) / 4r * d(n, 6)
  s -= 3 / 2r * d(n, 12)
  s -= 2 * d(n, 24)
  s += 4 * d(n, 42)
  s += 6 * d(n, 84)
  s += 6 * d(n, 120)
  (n * s).to_i
end

def a6(n)
  n * (4 * d(n, 30) - 4 * d(n, 60))
end

def a7(n)
  n * (d(n, 30) + 4 * d(n, 60))
end

def a(n, k)
  return a2(n) if k == 2
  return a3(n) if k == 3
  return a4(n) if k == 4
  return a5(n) if k == 5
  return a6(n) if k == 6
  return a7(n) if k == 7
end

def A(n)
  return [0] if n < 4
  return [0, a(n, 2)] if n % 2 == 1
  return [0] + (2..n / 2 - 1).map{|i| a(n, i)} + [1] if n / 2 < 9
  [0] + (2..7).map{|i| a(n, i)} + [0] * (n / 2 - 8) + [1]
end

n = 100
(1..n).each{|i|
  print i
  print ':'
  p A(i)
}

出力結果
1:[0]
2:[0]
3:[0]
4:[0, 1]
5:[0, 5]
6:[0, 12, 1]
7:[0, 35]
8:[0, 40, 8, 1]
9:[0, 126]
10:[0, 140, 20, 0, 1]
11:[0, 330]
12:[0, 228, 60, 12, 0, 1]
13:[0, 715]
14:[0, 644, 112, 0, 0, 0, 1]
15:[0, 1365]
16:[0, 1168, 208, 0, 0, 0, 0, 1]
17:[0, 2380]
18:[0, 1512, 216, 54, 54, 0, 0, 0, 1]
19:[0, 3876]
20:[0, 3360, 480, 0, 0, 0, 0, 0, 0, 1]
21:[0, 5985]
22:[0, 5280, 660, 0, 0, 0, 0, 0, 0, 0, 1]
23:[0, 8855]
24:[0, 6144, 864, 264, 24, 0, 0, 0, 0, 0, 0, 1]
25:[0, 12650]
26:[0, 11284, 1196, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
27:[0, 17550]
28:[0, 15680, 1568, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
29:[0, 23751]
30:[0, 13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1]
31:[0, 31465]
32:[0, 28448, 2464, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
33:[0, 40920]
34:[0, 37264, 2992, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
35:[0, 52360]
36:[0, 42444, 3924, 396, 216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
37:[0, 66045]
38:[0, 60648, 4332, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
39:[0, 82251]
40:[0, 75720, 5160, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
41:[0, 101270]
42:[0, 75012, 8148, 1134, 546, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
43:[0, 123410]
44:[0, 114400, 7040, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
45:[0, 148995]
46:[0, 138644, 8096, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
47:[0, 178365]
48:[0, 152064, 10560, 1200, 336, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
49:[0, 211876]
50:[0, 198200, 10600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
51:[0, 249900]
52:[0, 234208, 12064, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
53:[0, 292825]
54:[0, 254988, 15552, 1296, 648, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
55:[0, 341055]
56:[0, 321048, 15288, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
57:[0, 395010]
58:[0, 372708, 17052, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
59:[0, 455126]
60:[0, 375060, 25320, 3780, 720, 0, 300, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
61:[0, 521855]
62:[0, 494140, 21080, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
63:[0, 595665]
64:[0, 564800, 23360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
65:[0, 677040]
66:[0, 605352, 30360, 2310, 990, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
67:[0, 766480]
68:[0, 728960, 28288, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
69:[0, 864501]
70:[0, 823480, 30940, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
71:[0, 971635]
72:[0, 894816, 36288, 2520, 936, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
73:[0, 1088430]
74:[0, 1039404, 36852, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
75:[0, 1215450]
76:[0, 1161888, 40128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
77:[0, 1353275]
78:[0, 1241760, 50076, 3276, 1404, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
79:[0, 1502501]
80:[0, 1439440, 47120, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
81:[0, 1663740]
82:[0, 1595720, 50840, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
83:[0, 1837620]
84:[0, 1680588, 67620, 3612, 2352, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
85:[0, 2024785]
86:[0, 1945664, 58996, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
87:[0, 2225895]
88:[0, 2140600, 63448, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
89:[0, 2441626]
90:[0, 2270700, 77670, 4050, 1890, 360, 90, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
91:[0, 2672670]
92:[0, 2574528, 72864, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
93:[0, 2919735]
94:[0, 2814924, 77832, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
95:[0, 3183545]
96:[0, 2999040, 91008, 5088, 1824, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
97:[0, 3464840]
98:[0, 3345328, 88592, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
99:[0, 3764376]
100:[0, 3636800, 94400, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
投稿者 時刻: 0 件のコメント:
ラベル:

2024年6月22日土曜日

240622

PARI


Continued fraction for Pi^n(2)

Pi^3の連分数展開も興味深い。

for(n=1, 10, print([n, Pi^n, contfrac(Pi^n)]))

出力結果
[1, 3.1415926535897932384626433832795028842, [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6]]
[2, 9.8696044010893586188344909998761511353, [9, 1, 6, 1, 2, 47, 1, 8, 1, 1, 2, 2, 1, 1, 8, 3, 1, 10, 5, 1, 3, 1, 2, 1, 1, 3, 15, 1, 1, 2, 2, 1, 3, 2, 7, 1, 9, 18]]
[3, 31.006276680299820175476315067101395202, [31, 159, 3, 7, 1, 13, 2, 1, 3, 1, 12, 2, 2, 4, 34, 2, 43, 3, 1, 3, 2, 1, 1, 5, 1, 1, 4, 1, 5, 4, 2, 4]]
[4, 97.409091034002437236440332688705111250, [97, 2, 2, 3, 1, 16539, 1, 6, 7, 6, 8, 6, 3, 9, 1, 1, 1, 18, 1, 4, 1, 13, 1, 2, 1, 127, 2]]
[5, 306.01968478528145326274131004343560648, [306, 50, 1, 4, 60, 1, 4, 23, 1, 2, 1, 1, 1, 17, 1, 5, 1, 1, 1, 2, 4, 10, 4, 1, 3, 2, 1, 36, 7, 1, 27]]
[6, 961.38919357530443703021944365241989886, [961, 2, 1, 1, 3, 9, 1, 5, 1, 3, 6, 1, 3, 1, 77, 11, 3, 1, 11, 14, 4, 4, 27, 5, 178, 6]]
[7, 3020.2932277767920675142064930720418319, [3020, 3, 2, 2, 3, 2, 10, 4, 1, 1, 2, 1, 3, 13, 58, 2, 50, 1, 2, 21, 1, 1, 2, 1, 14, 1, 182, 2, 1, 1, 2]]
[8, 9488.5310160705740071285755039067657967, [9488, 1, 1, 7, 1, 1, 3, 1, 1, 1, 4, 10, 13, 1, 1, 2, 3, 6, 1, 1, 1, 135, 1, 2, 2, 3, 1, 2, 1, 7, 3, 1, 1, 4, 2, 7, 1, 2]]
[9, 29809.099333446211666509402401239655364, [29809, 10, 14, 1, 9, 3, 1, 5, 8, 16, 3, 15, 7, 39, 3, 3, 1, 1, 2, 1, 3, 4, 1, 1, 7, 2, 1, 2, 1, 3]]
[10, 93648.047476083020973716690184919345635, [93648, 21, 15, 1, 4, 2, 1, 3, 2, 1, 4, 2, 74, 26, 5, 3, 1, 5, 17, 1, 2, 1, 1, 2, 1, 14, 2, 2, 8]]
投稿者 時刻: 0 件のコメント:

2024年5月27日月曜日

240527

Ruby


GCD sum theorems

次の論文のTheorem 3 を確認してみた。
GCD sum theorems. Two Multivariable Cesaro Type Identities

class Array
  def gcd
    self.inject{|a, b| a.gcd(b)}
  end
end

def A(k, m, n)
  s = 0
  (1..n).to_a.repeated_permutation(k){|perm|
    s += (perm + [n]).gcd ** m
  }
  s
end

n = 15
# i>=j
(1..4).each{|i|
  (1..i).each{|j|
    a = (1..n).map{|k| A(i, j, k)}
    b = (1..n).map{|k| A(j, i, k)}
    p [i, j, a] if a == b
  }
}

出力結果
[1, 1, [1, 3, 5, 8, 9, 15, 13, 20, 21, 27, 21, 40, 25, 39, 45]]
[2, 1, [1, 5, 11, 22, 29, 55, 55, 92, 105, 145, 131, 242, 181, 275, 319]]
[2, 2, [1, 7, 17, 40, 49, 119, 97, 208, 225, 343, 241, 680, 337, 679, 833]]
[3, 1, [1, 9, 29, 74, 129, 261, 349, 596, 789, 1161, 1341, 2146, 2209, 3141, 3741]]
[3, 2, [1, 11, 35, 100, 149, 385, 391, 848, 1017, 1639, 1451, 3500, 2365, 4301, 5215]]
[3, 3, [1, 15, 53, 176, 249, 795, 685, 1856, 2133, 3735, 2661, 9328, 4393, 10275, 13197]]
[4, 1, [1, 17, 83, 274, 629, 1411, 2407, 4388, 6729, 10693, 14651, 22742, 28573, 40919, 52207]]
[4, 2, [1, 19, 89, 316, 649, 1691, 2449, 5104, 7281, 12331, 14761, 28124, 28729, 46531, 57761]]
[4, 3, [1, 23, 107, 424, 749, 2461, 2743, 7232, 9369, 17227, 15971, 45368, 30757, 63089, 80143]]
[4, 4, [1, 31, 161, 736, 1249, 4991, 4801, 15616, 19521, 38719, 29281, 118496, 57121, 148831, 201089]]
投稿者 時刻: 0 件のコメント:
ラベル:

2024年4月14日日曜日

240414

PARI


Apéry numberの仲間たち

l>=7 のとき、closed formはまだ見つかっていないようだ。

h(n) = sum(k=1, n, 1/k);
a(n, l) = sum(k=0, n, binomial(n, k)^l*(1-l*k*(h(k)-h(n-k))));
for(l=1, 12, for(n=0, 10, print1(a(n, l), ", ")); print)

出力結果
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
1, -2, 6, -20, 70, -252, 924, -3432, 12870, -48620, 184756, 
1, -3, 19, -147, 1251, -11253, 104959, -1004307, 9793891, -96918753, 970336269, 
1, -4, 48, -760, 13840, -273504, 5703096, -123519792, 2751843600, -62659854400, 1451780950048, 
1, -5, 109, -3317, 121501, -4954505, 216867925, -9981053045, 476860000285, -23451310381505, 1180189308268609, 
1, -6, 234, -13164, 936810, -76041756, 6739851636, -635746866264, 62819253878634, -6434646802174140, 678284981981948484, 
1, -7, 487, -49255, 6669751, -1053222757, 184786011019, -34916924925799, 6974335808409655, -1454249641139993257, 313810538382057898237, 
1, -8, 996, -177200, 45086980, -13631185008, 4662729892464, -1737911032294368, 690950882254608900, -288779498502632656640, 125605455885282444138496, 
1, -9, 2017, -620073, 294290001, -168314616009, 111074551176961, -80766140154871593, 63220243361206024657, -52382852763175965870009, 45422849668460020450930017, 
1, -10, 4062, -2125828, 1874792350, -2009033203260, 2538914209085100, -3571944365364382920, 5460901555410763324830, -8898003584839672562442700, 15258440639018489319278045812,
投稿者 時刻: 0 件のコメント:
ラベル:

2024年3月30日土曜日

240330

PARI


Continued fraction for Pi^n(1)

Pi^4の連分数展開が興味深い。

default(realprecision, 70);
for(n=1, 100, print([n, contfrac(Pi^n)]))

出力結果
[1, [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4]]
[2, [9, 1, 6, 1, 2, 47, 1, 8, 1, 1, 2, 2, 1, 1, 8, 3, 1, 10, 5, 1, 3, 1, 2, 1, 1, 3, 15, 1, 1, 2, 2, 1, 3, 2, 7, 1, 9, 18, 30, 2, 145, 1, 1, 17, 9, 1, 1, 1, 1, 7, 12, 1, 2, 1, 12, 1, 1, 4, 1, 5, 1, 1, 2, 3, 4, 1, 3, 2, 9, 1, 20, 11, 14]]
[3, [31, 159, 3, 7, 1, 13, 2, 1, 3, 1, 12, 2, 2, 4, 34, 2, 43, 3, 1, 3, 2, 1, 1, 5, 1, 1, 4, 1, 5, 4, 2, 4, 11, 3, 3, 1, 1, 2, 1, 7, 2, 1, 1, 3, 1, 12, 3, 1, 9, 2, 1, 8, 23, 1, 45, 1, 1, 2, 1, 23, 3, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 16]]
[4, [97, 2, 2, 3, 1, 16539, 1, 6, 7, 6, 8, 6, 3, 9, 1, 1, 1, 18, 1, 4, 1, 13, 1, 2, 1, 127, 1, 1, 1, 4, 1, 6, 1, 1, 1, 10, 10, 1, 1, 2, 1, 2, 1, 5, 1, 1, 10, 1, 3, 2, 1, 1, 4, 9, 1, 7, 70, 1, 13, 1, 2, 6, 1, 2, 24, 5, 2]]
[5, [306, 50, 1, 4, 60, 1, 4, 23, 1, 2, 1, 1, 1, 17, 1, 5, 1, 1, 1, 2, 4, 10, 4, 1, 3, 2, 1, 36, 7, 1, 27, 3, 1, 6, 1, 2, 1, 1, 3, 1, 1, 1, 53, 1, 1, 1, 15, 1, 3, 573, 1, 5, 2, 1, 4, 1, 3, 2, 1, 15, 2, 1, 1, 1, 4, 3, 1, 66]]
[6, [961, 2, 1, 1, 3, 9, 1, 5, 1, 3, 6, 1, 3, 1, 77, 11, 3, 1, 11, 14, 4, 4, 27, 5, 178, 5, 1, 2, 2, 1099, 2, 3, 71, 4, 1, 3, 1, 1, 1, 2, 16, 154, 1, 3, 1, 5, 4, 5, 1, 1, 17, 1, 1, 1, 1, 1, 2, 4]]
[7, [3020, 3, 2, 2, 3, 2, 10, 4, 1, 1, 2, 1, 3, 13, 58, 2, 50, 1, 2, 21, 1, 1, 2, 1, 14, 1, 182, 2, 1, 1, 2, 69, 6, 2, 4, 3, 2, 5, 11, 2, 2, 5, 4, 1, 5, 41, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 2, 3]]
[8, [9488, 1, 1, 7, 1, 1, 3, 1, 1, 1, 4, 10, 13, 1, 1, 2, 3, 6, 1, 1, 1, 135, 1, 2, 2, 3, 1, 2, 1, 7, 3, 1, 1, 4, 2, 7, 1, 1, 1, 27, 4, 1, 3, 13, 1, 3, 2, 17, 1, 1, 3, 3, 2, 1, 1, 1, 1, 1, 6, 5, 5, 5, 1, 2, 1, 1, 1, 2, 2, 2, 1, 6, 11, 2, 24, 3, 25, 3]]
[9, [29809, 10, 14, 1, 9, 3, 1, 5, 8, 16, 3, 15, 7, 39, 3, 3, 1, 1, 2, 1, 3, 4, 1, 1, 7, 2, 1, 2, 1, 3, 10, 5, 72, 1, 1, 3, 2, 1, 2, 2, 6, 3, 7, 2, 1, 3, 6, 1, 6, 2, 3, 20, 1, 2, 8, 9, 1, 14, 3, 1, 2, 4, 14]]
[10, [93648, 21, 15, 1, 4, 2, 1, 3, 2, 1, 4, 2, 74, 26, 5, 3, 1, 5, 17, 1, 2, 1, 1, 2, 1, 14, 2, 2, 10, 7, 4, 2, 2, 3, 1, 2, 1, 3, 3, 1, 1, 1, 23, 13, 2, 2, 1, 1, 2, 13, 9, 9, 1, 6, 1, 2, 1, 7, 2, 2, 12, 6, 1, 16, 8]]
[11, [294204, 55, 1, 1, 1, 2, 1, 76, 1, 10, 1, 1, 1, 1, 10, 1, 41, 3, 1, 1, 2, 2, 3, 6, 6, 2, 1, 5, 1, 1, 1, 12, 1, 61, 6, 2, 44, 1, 8, 1, 5, 5, 1, 2, 1, 5, 3, 1, 2, 8, 4, 1, 2, 3, 2, 2, 1, 1, 3, 1, 21, 2, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 1, 6]]
[12, [924269, 5, 1, 1, 27, 2, 20, 3, 1, 15, 3, 1, 1, 2, 2, 1, 5, 4, 3, 2, 1, 1, 6, 64, 9, 5, 3, 1, 8, 1, 1, 2, 1, 2, 8, 1, 2, 1, 1, 1, 2, 4, 1, 1, 26, 1, 50, 2, 1, 3, 1, 1, 19, 1, 4, 3, 6, 1, 1, 1, 15, 1, 3, 1, 1, 2, 1, 1, 7, 5, 1, 1, 1, 1, 5]]
[13, [2903677, 3, 1, 2, 3, 1, 1, 4, 1, 24, 1, 1, 1, 2, 1, 26, 1, 4, 2, 1, 5, 1, 1, 6, 1, 7, 22, 2, 1, 7, 5, 1, 4, 1, 1, 1, 1, 1, 6, 1, 1, 1, 5, 1, 2, 1, 2, 146, 2, 1, 1, 2, 2, 3, 5, 3, 2, 1, 1, 1, 24, 2, 2, 3, 2, 14, 1, 2, 1, 5, 11, 4, 3, 2, 1, 1, 5]]
[14, [9122171, 5, 1, 1, 128, 1, 14, 41, 2, 2, 1, 1, 1, 3, 1, 9, 4, 1, 1, 1, 144, 2, 1, 14, 3, 2, 10, 1, 1, 2044, 1, 2, 2, 2, 1, 3, 5, 1, 1, 1, 1, 1, 5, 404, 1, 1, 1, 1, 16, 1, 2, 21, 1, 1, 3, 46, 10, 12]]
[15, [28658145, 1, 31, 1, 2, 427, 1, 1, 6, 1, 1, 5, 1, 1, 4, 3, 14, 5, 7, 2, 1, 4, 1, 8, 6, 1, 2, 1, 1, 2, 1, 6, 15, 8, 21, 1, 1, 1, 11, 2, 2, 3, 1, 3, 2, 1, 3, 101, 1, 2, 3, 5, 1, 1, 10, 45, 4, 1, 4, 1, 3, 10, 3]]
[16, [90032220, 1, 5, 2, 1, 2, 1, 1, 1, 19, 1, 3, 1, 2, 7, 2, 2, 1, 1, 2, 8, 2, 1, 4, 17, 1, 7, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 12, 18, 1, 2, 4, 2, 5, 6, 1, 14, 1, 1, 1, 13, 1, 9, 1, 1, 33, 1, 1, 1, 2, 2, 2, 21, 3, 5, 3, 1, 1, 1, 4, 1, 2, 1, 1, 2, 4, 4]]
[17, [282844563, 1, 1, 2, 2, 1, 1, 3, 16, 1, 4, 2, 3, 2, 1, 2, 13, 2, 2, 6, 12, 5, 1, 3, 1, 1, 6, 1, 1, 1, 1, 3, 4, 3, 2, 2, 1, 3, 4, 2, 3, 1, 6, 15, 1, 2, 1, 1, 1, 13, 1, 1, 1, 7, 2, 20, 2, 1, 2, 2, 9, 3, 1, 2, 6, 1, 1, 3, 5, 1, 21, 1, 4, 3, 1, 4]]
[18, [888582403, 14, 30, 1, 4, 2, 1, 1, 2, 5, 152, 3, 1, 6, 5, 7, 2, 1, 9, 1, 1, 1, 267, 1, 1, 3, 37, 1, 5, 1, 1, 4, 1, 2, 2, 9, 1, 4, 4, 70, 1, 2, 1, 3, 1, 66, 4, 1, 2, 3, 1, 8, 2, 7, 1, 2, 9, 3, 6]]
[19, [2791563949, 1, 1, 2, 18, 6, 89, 163, 1, 68, 5, 1, 1, 2, 1, 4, 29, 13, 1, 1, 1, 1, 1, 3, 1, 2, 2, 70, 4, 1, 3, 6, 1, 2, 1, 4, 1, 3, 5, 5, 1, 2, 1, 1, 3, 3, 1, 3, 1, 6, 1, 1, 1, 1, 3, 5, 2, 1, 3, 2, 2, 1, 1, 1, 2, 6, 6]]
[20, [8769956796, 12, 10, 1, 6, 1, 5, 2, 1, 3, 1, 93, 1, 1, 2, 1, 12, 4, 3, 4, 5, 2, 2, 3, 6, 1, 1, 2, 14, 1, 14, 1, 1, 2, 2, 12, 44, 1, 1, 1, 6, 1, 1, 1, 4, 1, 3, 3, 4, 4, 2, 1, 4, 1, 9, 1, 3, 7, 6, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 2]]
[21, [27551631842, 1, 6, 1, 8, 3, 1, 7, 1, 1, 1, 2, 9, 1, 23, 1, 2, 649, 2, 2, 1, 8, 1, 2, 6, 1, 1, 1, 153, 2, 2, 743, 1, 22, 1, 2, 1, 11, 1, 3, 11, 2, 1, 1, 3, 1, 1, 1, 6, 1, 3, 4, 5, 1, 3, 2, 1, 2, 1, 16, 12]]
[22, [86556004191, 1, 52, 1, 1, 2, 7, 1, 1, 36, 1, 7, 2, 1, 57, 18, 11, 401, 2, 46, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 1, 2, 19, 23, 2, 2, 1, 9, 4, 1, 1, 2, 2, 2, 7, 3, 3, 1, 2, 1, 34, 5, 1, 1, 4, 25]]
[23, [271923706893, 1, 1, 1, 1, 1, 1, 10, 4, 3, 2, 2, 3, 1, 26, 163, 1, 80, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 12, 2, 1, 1, 5, 1, 2, 10, 3, 2, 1, 5, 1, 16, 1, 51, 4, 1, 2, 2, 1, 8, 1, 1, 6, 2, 2, 1, 1, 1, 1, 1, 2, 3, 14, 1, 2, 1, 1, 29]]
[24, [854273519913, 1, 7, 1, 13, 2, 1, 4, 2, 1, 44, 1, 1, 4, 2, 1, 2, 4, 1, 1, 1, 6, 1, 4, 1, 1, 3, 1, 1, 1, 5, 3, 3, 31, 1, 1, 1, 5, 2, 1, 2, 1, 26, 8, 1, 2, 2, 8, 1, 7, 2, 1, 1, 1, 1, 1, 1, 1, 6, 2, 1, 23, 5, 2, 3, 1, 14, 1, 9]]
[25, [2683779414317, 1, 3, 4, 21, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 6, 1, 49, 1, 1, 13, 3, 1, 4, 1, 3, 7, 2, 8, 1, 5, 2, 1, 2, 3, 7, 1, 1, 2, 1, 1, 8, 1, 2, 1, 11, 1, 12, 1, 2, 1, 4, 9, 3, 1, 2, 19, 1, 1, 7, 1, 2, 2, 1, 1, 1, 4, 3, 2, 1, 5, 1, 1, 3]]
[26, [8431341691876, 4, 1, 4, 1, 6, 6, 7, 1, 2, 26, 1, 3, 1, 7, 4, 4, 1, 1, 1, 1, 1, 3, 1, 2, 4, 1, 1, 14, 65, 2, 3, 12, 1, 1, 3, 3, 2, 11, 5, 11, 1, 1, 4, 2, 2, 1, 9, 1, 1, 1, 1, 1, 18, 1, 1, 1, 2, 2, 3, 1, 2, 2, 1, 5, 1, 2, 2, 1, 1, 2]]
[27, [26487841119103, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 4, 1, 2, 2, 6, 2, 97, 1, 1, 7, 1, 1, 1, 1, 1, 3, 5, 1, 2, 1, 9, 1, 2, 1, 1, 5, 3, 57, 1, 4, 2, 11, 1, 1, 16, 6, 2, 2, 3, 1, 1, 1, 5, 20, 9, 4, 6, 9, 1, 1, 1, 3, 13, 1, 3, 7]]
[28, [83214007069229, 1, 1, 1, 1, 2, 1, 1, 1, 25, 1, 4, 1, 4, 1, 34, 17, 2, 1, 3, 1, 4, 1, 3, 5, 1, 5, 6, 4, 11, 3, 23, 1, 2, 3, 2, 5, 1, 1, 1, 6, 3, 3, 1, 1, 13, 1, 1, 10, 1, 16, 1, 1, 906, 1, 2, 27, 36]]
[29, [261424513284460, 1, 6, 1, 2, 1, 212, 1, 19, 1, 3, 1, 169, 2, 1, 1, 1, 26, 5, 6, 2, 10, 1, 3, 2, 4, 1, 8, 1, 3, 5, 21, 5, 2, 1, 1, 9, 1, 2, 1, 1, 1, 4, 2, 9, 1, 5, 18, 2, 2, 1, 1, 6, 1, 8, 1, 3, 3, 12]]
[30, [821289330402749, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 2, 10, 1, 2, 1, 1, 1, 3, 5, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 6, 6, 2, 20, 11, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 4, 1, 3, 5, 6, 1, 1, 1, 2, 4, 2, 1, 8, 1, 6, 1, 1, 137, 7, 1, 3, 5, 1, 1, 2, 39, 3]]
[31, [2580156526864958, 1, 1, 23, 1, 1, 8, 12, 3, 4, 433, 1, 2, 13, 1, 1, 2, 1, 30, 2, 1, 1, 1, 5, 8, 11, 2, 1, 3, 7, 1, 9, 180, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 10, 2, 2, 1, 1, 1, 2, 2, 15, 649]]
[32, [8105800789910709, 1, 1, 1, 7, 1, 1, 3, 1, 8, 2, 2, 1, 3, 3, 2, 9, 2, 3, 99, 35, 5, 4, 3, 7, 1, 4, 1, 17, 1, 1, 1, 2, 2, 1, 1, 1, 10, 5, 70, 1, 40, 1, 2, 10, 7, 2, 2, 1, 1, 49, 1, 3, 16]]
[33, [25465124213045828, 2, 7, 1, 736, 3, 1, 5, 1, 35, 1, 12, 5, 3, 6, 3, 11, 2, 15, 1, 1, 2, 2, 2, 9, 2, 1, 4744, 4, 12, 3, 2, 15, 1, 1, 1, 200, 10, 1, 1, 3, 4, 6, 1, 2]]
[34, [80001047150456339, 1, 1, 4, 4, 1, 1, 16, 1, 1, 2, 2, 1, 1, 4, 37, 7, 1, 2, 1, 5, 1, 1, 2, 1, 24, 1, 8, 1, 1, 1, 1, 13, 2, 1, 13, 1, 3, 1, 3, 1, 2, 38, 3, 3, 1, 1, 2, 1, 2, 1, 4, 1, 1, 3, 2, 1, 124, 1, 11, 1, 2, 1, 12, 2, 2]]
[35, [251330702007364298, 1, 1, 1, 1, 1, 1, 7, 1, 3, 1, 1, 1, 5834, 1, 3, 34, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 1, 7, 63, 3, 6, 3, 1, 6, 1, 12, 6, 1, 2, 2, 15, 1, 6, 4, 2, 5, 1, 10, 1, 5, 18, 13, 1, 4]]
[36, [789578687047901181, 11, 3, 1, 6, 37, 6, 1, 6, 1, 2, 3, 3, 2, 2, 1, 3, 10, 2, 2, 3, 2, 1, 1, 4, 90, 1, 2, 1, 3, 1, 1, 28, 40, 1, 1, 3, 2, 5, 1, 4, 37, 1, 12, 3, 9, 5, 1, 4, 1, 1, 2, 1, 1, 6]]
[37, [2480534602660760780, 2, 3, 3, 1, 2, 5, 1, 1, 1, 5, 1, 2, 1, 64, 7, 1, 1, 7, 1, 1, 16, 1, 84, 1, 2, 1, 1, 3, 4, 1228, 13, 2, 1, 13, 1, 1, 2, 4, 1, 1, 10, 4, 5, 2, 3, 1, 3, 1, 2, 2, 1, 2, 1, 2, 7, 2]]
[38, [7792829284694322855, 1, 1, 1, 1, 1, 7, 3, 3, 141, 1, 2, 2, 2, 1, 1, 192, 1, 5, 1, 1, 3, 1, 8, 14, 1, 2, 49, 1, 12, 1, 4, 2, 1, 12, 2, 14, 12, 246, 1, 35, 1, 1, 3, 6, 1, 1, 1, 1, 14, 4]]
[39, [24481895231475088054, 1, 1, 1, 3, 1, 3, 1, 1, 7, 1, 1, 3, 42, 1, 150, 1, 2, 1, 1, 5, 1, 1, 1, 8, 2, 9, 1, 2, 1, 2, 2, 15, 2, 1, 1, 5, 1, 1, 1, 1, 4, 4, 2, 2, 1, 17, 3, 2, 1, 1, 2, 5, 24, 2, 8, 2, 1, 1, 1, 7, 1, 2]]
[40, [76912142205157127257, 3, 1, 3, 2, 1, 2, 1, 4, 1, 7, 79, 2, 1, 3, 2, 2, 1, 1, 5, 7, 31, 1, 1, 4, 1, 1, 15, 1, 2, 1, 1, 18, 1, 8, 1, 7, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 6, 4, 1, 7, 20, 1, 1, 3, 1, 1, 1, 3, 15, 2]]
[41, [241626620923575111130, 3, 2, 5, 4, 1, 2, 7, 1, 4, 8, 1, 18, 1, 5, 3, 1, 3, 1, 1, 2, 3, 4, 10, 1, 1, 10, 1, 2, 1, 4, 37, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 4, 1, 2, 3, 5, 1, 1, 110, 1, 1, 2, 3, 1, 4, 1, 9, 1, 1, 2]]
[42, [759092417205229390873, 1, 7, 11, 2, 1, 6, 2, 2, 1, 1, 2, 1, 2, 104, 25, 13, 6, 1, 3, 3, 2, 1, 7, 2, 2, 4, 2, 1, 5, 2, 3, 1, 1, 1, 1, 3, 1, 1, 1, 16, 5, 1, 18, 40, 2, 1, 2, 1, 1, 16, 4, 11]]
[43, [2384759161287667022584, 1, 16, 1, 2, 4, 1, 5, 1, 1, 12, 3, 1, 3, 178, 1, 23, 1, 5, 2, 1, 1, 4, 1, 1, 6, 1, 1, 2, 1, 4, 1, 14, 3, 1, 1, 10, 10, 1, 2, 12, 2, 1, 115, 1, 885, 1, 1, 2]]
[44, [7491941861682291566352, 1, 1, 4, 7, 5, 1, 1, 1, 5, 2, 1, 1, 5, 6, 1, 18, 2, 4, 139, 1, 25, 4, 2, 1, 1, 2, 86, 1, 3, 3, 1, 4, 3, 5, 1, 2, 10, 31, 2, 1, 3, 5, 14, 1, 2, 1, 1, 1, 1, 1, 1, 11]]
[45, [23536629513782926058032, 2, 23, 3, 4, 4, 2, 1, 2, 1, 1, 5, 1, 4, 2, 1, 1, 1, 1, 2, 2, 5, 1, 6, 2, 4, 2, 1, 5, 2, 30, 1, 1, 1, 3, 1, 39, 2, 5, 3, 4, 7, 1, 1, 40, 3, 12, 2, 1, 1, 2, 1, 8, 1, 1, 1, 15]]
[46, [73942502370765147684186, 1, 1, 2, 2, 8, 7, 1, 2, 2, 1, 16, 1, 1, 1, 1, 6, 1, 2, 1, 9, 20, 7, 1, 9, 10, 5, 9, 1, 1, 16, 1, 2, 1, 8, 1, 2, 1, 3, 10, 2, 1, 1, 1, 1, 4, 1, 1, 301, 1, 3, 1, 10]]
[47, [232297222236041657886385, 1, 3, 1, 77, 2, 13, 1, 1, 5, 3, 1, 3, 2, 24, 51, 25, 1, 1, 2, 6, 13, 4, 1, 1, 13, 3, 1, 1, 8, 1, 41, 3, 9, 13, 2, 7, 1, 1, 1, 14, 1, 1, 1, 5, 1, 4]]
[48, [729783246826064035206280, 1, 1, 12, 1, 2, 2, 2, 1, 10, 8, 2, 1, 1, 9, 5, 3, 8, 2, 1, 38, 3, 1, 4, 1, 20, 1, 1, 1, 16, 1, 2, 1, 7, 5, 1, 1, 10, 1, 1, 2, 6, 34, 4, 1, 1, 2, 3, 1, 2, 1, 13, 1, 1, 2, 2]]
[49, [2292681686941669566432911, 7, 4, 3, 1, 82, 5, 13, 3, 3, 1, 3, 1, 3, 1, 2, 1, 1, 5, 1, 11, 1, 110, 1, 11, 1, 1, 1, 3, 3, 1, 1, 1, 31, 1, 4, 1, 3, 7, 3, 14, 1, 2, 1, 4, 170, 3, 3]]
[50, [7202671944715803306364652, 1, 2, 19, 4, 1, 2, 1, 1, 116, 4, 1, 1, 1, 2, 2, 1, 7, 1, 3, 2, 8, 1, 2, 2, 9, 4, 9, 2, 3, 7, 1, 1, 1, 1, 7, 2, 1, 4, 11, 14, 1, 2, 2, 47, 3, 1, 1, 3, 2, 5]]
[51, [22627861267736477052126260, 8, 1, 4, 1, 1, 1, 3, 1, 1, 1, 215, 8, 170, 3, 2, 4, 1, 2, 7, 2, 1, 9, 1, 3, 1, 5, 2, 1, 3, 2, 3, 7, 1, 1, 1, 3, 3, 27, 1, 1, 1, 1, 9, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 3]]
[52, [71087522725169941823644668, 1, 4, 1, 2, 1, 1, 33, 2, 2, 2, 1, 1, 1, 8, 1, 2, 1, 40, 4, 4, 2, 1, 1, 3, 28, 1, 1, 1, 26, 1, 1, 29, 3, 2, 150, 1, 7, 3, 104, 4, 4, 2, 3, 1, 19]]
[53, [223328039155291367651964249, 2, 1, 2, 20, 1, 10, 1, 3, 1, 27, 5, 1, 1, 1, 6, 2, 3, 1, 2, 7, 3, 1, 1, 1, 3, 9, 1, 1, 2, 2, 1, 3, 2, 2, 1, 1, 1, 5, 3, 3, 3, 258, 4, 7, 1, 1, 2, 2, 2, 5, 1, 5, 2]]
[54, [701605727150877054142644067, 33, 1, 1, 1, 2, 3, 4, 8, 21, 1, 4, 3, 2, 9, 271, 34, 1, 2, 1, 6, 2, 2, 6, 1, 2, 3, 1, 43, 61, 1, 1, 2, 1, 21, 1, 39, 2, 2, 3]]
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[56, [6924570972517795883320726281, 1, 1, 4, 1, 2, 10, 5, 1, 1, 3, 1, 5, 5, 5, 2, 12, 29, 1, 5, 1, 70, 8, 1, 3, 1, 1, 4, 3, 10, 2, 2, 17, 1, 5, 1, 10, 4, 2, 3, 1, 3, 1, 1, 1, 1, 3, 3]]
[57, [21754181296523037597639725717, 1, 6, 1, 1, 4, 3, 4, 1, 2, 3, 2, 3, 22, 1, 2, 2, 1, 1, 2, 2, 1, 4, 15, 1, 63, 6, 5, 3, 2, 17, 5, 1, 29, 1, 2, 3, 2, 3, 1, 1, 1, 4, 1, 2, 1, 2, 10]]
[58, [68342776146017258398967520977, 1, 188, 1, 2, 2, 1, 2, 2, 4, 3, 8, 1, 3, 3, 2, 1, 2, 3, 2, 4, 1, 3, 14, 1, 1, 2, 2, 5, 1, 1, 1, 1, 3, 1, 5, 1, 13, 1, 1, 1, 3, 14, 3, 15, 1, 4, 2, 4, 1, 1, 1, 1, 2]]
[59, [214705163466259581466085919504, 117, 1, 2, 1, 3, 1, 3, 4, 3, 9, 2, 1, 2, 4, 1, 1, 7, 3, 1, 2, 1, 2, 4, 2, 1, 2, 45, 1, 92, 2, 4, 3, 114, 5, 1, 29, 1, 1, 1, 1, 3, 11]]
[60, [674516164233396768200126958915, 1, 1, 3, 1, 2, 1, 1, 1, 131, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 8, 2, 1, 5, 1, 1, 17, 5, 10, 1, 3, 1, 1, 4, 2, 3, 1, 1, 1, 16, 1, 4, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 13, 1, 25]]
[61, [2119055026283205737110078195749, 2, 36, 1, 1, 5, 1, 2, 1, 4, 2, 7, 1, 1, 9, 2, 2, 1, 71, 17, 1, 45, 4, 1, 1, 1, 26, 2, 3, 6, 3, 18, 2, 1, 3, 3, 1, 7, 1, 2, 1, 4, 3]]
[62, [6657207703123845367424584915447, 1, 1, 7, 2, 1, 3, 1, 8, 34, 1, 18, 3, 4, 2, 1, 1, 6, 3, 2, 1, 389, 1, 2, 21, 4, 7, 9, 1, 6, 1, 1, 1, 6, 1, 5, 6, 77]]
[63, [20914234813555253845753030071176, 1, 4, 1, 1, 1, 1, 1, 2, 2, 2, 1, 5, 1, 2, 2, 1, 1, 1, 1, 1, 71, 20, 1, 1, 1, 1, 6, 1, 1, 2, 1, 18, 6, 1, 10, 1, 6, 4, 1, 1, 2, 1, 52, 3, 3, 2, 5, 2, 7, 1, 2]]
[64, [65704006445717084572022626334540, 1, 31, 1, 1, 11, 1, 5, 11, 1, 47, 1, 5, 1, 1, 2, 28, 10, 1, 3, 2, 7, 1, 5, 1, 6, 5, 1, 11, 5, 3, 1, 5, 2, 1, 2, 2, 5, 2, 2, 1, 2, 1, 13]]
[65, [206415223961281214949635806621076, 8, 1, 2, 15, 2, 1, 3, 1, 2, 2, 1, 1, 1, 7, 1, 1, 7, 3, 3, 1, 1, 1, 5, 1, 20, 1, 3, 392, 1, 1, 2, 3, 2, 1, 4, 2, 1, 2, 17, 1, 25, 1, 6, 7]]
[66, [648472551185852924760805487074064, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 33, 1, 46, 1, 2, 3, 1, 1, 1, 4, 3, 2, 3, 4, 3, 1, 35, 1, 1, 1, 3, 22, 1, 3, 2, 14, 19, 1, 1, 1, 12, 8, 4]]
[67, [2037236602860106711985144771884927, 2, 1, 1, 5, 2, 1, 1, 1, 1, 1, 13, 1, 24, 1, 1, 1, 1, 5, 1, 4, 3, 2, 1, 1, 19, 1, 1, 2, 2, 1, 14, 1, 6, 2, 1, 2, 6, 1, 3, 2, 1, 6, 4, 3, 1, 7, 16, 3, 1, 1, 2]]
[68, [6400167545169538406684014293670341, 1, 1, 4, 10, 1, 6, 1, 1, 3, 1, 1, 4, 5, 1, 18, 1, 3, 1, 5, 1, 2, 7, 2, 4, 1, 1, 1, 1, 2, 6, 1, 2, 2, 1, 6, 4, 1, 1, 1, 18, 1, 4, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 17]]
[69, [20106719341648443041008873880038160, 1, 10, 11, 1, 54, 1, 2, 57, 2, 2, 1, 1, 10, 1, 1, 4, 9, 1, 1, 1, 1, 1, 1, 5, 1, 5, 4, 1, 5, 7, 6, 50, 1, 24, 1, 2, 6]]
[70, [63167121771514552681892760031018334, 1, 2, 1, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 2, 4, 51, 2, 2, 1, 2, 2, 1, 4, 1, 3, 2, 7, 3, 24, 539, 8, 1, 2, 2, 9, 1, 1, 5, 1, 1, 10]]
[71, [198445365705802004702001456166116205, 1, 3, 2, 34, 1, 4, 1, 5, 1, 1, 1, 10, 1, 1, 1, 1, 1, 10, 7, 36, 3, 6, 3, 1, 1, 2, 4, 4, 2, 1, 4, 11, 294, 1, 56, 2]]
[72, [623434503040287472342007784213127357, 1, 3, 1, 1, 2, 1, 6, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 5, 3, 4, 1, 1, 7, 44, 1, 1, 2, 13, 1, 97, 1, 3, 5, 1, 1, 1, 2, 1, 4, 2]]
[73, [1958577254745770740635072198655932631, 27, 1, 1, 1, 4, 1, 4, 35, 7, 1, 19, 1, 1, 5, 2, 4, 3, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 5, 8, 1, 1, 1, 1, 2, 41, 1, 1, 1, 48, 1, 19]]
[74, [6153051914997378363457293172896716008, 1, 18, 1, 4, 23, 2, 1, 15, 1, 2, 14, 1, 4, 3, 1, 13, 1, 3, 1, 1, 3, 4, 6, 1, 4, 2, 4, 2, 7, 4, 4, 5, 2, 9, 3, 2]]
[75, [19330382693312372796273669213182810875, 1, 3, 6, 1, 4, 2, 2, 11, 1, 2, 1, 1, 1, 1, 2, 3, 33, 1, 1, 1, 2, 2, 4, 1, 10, 1, 1, 5, 47, 13, 5987]]
[76, [60728188260389431620181363923786356434, 5, 16, 4, 3, 1, 3, 1, 4, 6, 1, 4, 12, 1, 5, 2, 2, 4, 1, 1, 7, 3, 1, 3, 3, 1, 1, 1, 2, 13, 1, 1, 1, 2, 1, 2, 1, 36, 9, 2, 4]]
[77, [190783230104657364116871823923953977404, 4, 4, 4, 15, 2, 2, 4, 1, 5, 1, 24, 1, 6, 2, 3, 3, 1, 11, 1, 4, 2, 2, 1, 3, 1, 2, 2, 1, 10, 1, 3, 6, 2, 36, 2, 4, 3]]
[78, [599363194124922655319699900093848879647, 1, 17, 1, 13, 1, 2, 6, 5, 4, 1, 8, 1, 1, 42, 2, 31, 1, 26, 7, 5, 1, 2, 1, 2, 1, 20, 1, 23, 5, 1, 5, 4]]
[79, [1882955007494970137423872697623397063368, 1, 8, 3, 2, 1, 3, 1, 2, 6, 2, 1, 6, 3, 3, 4, 1, 1, 1, 1, 3, 5, 2, 2, 2, 1, 4, 1, 4, 1, 1, 1, 1, 1, 18, 1, 6, 1, 3, 4, 1, 76]]
[80, [5915477618586312249935148194956040327282, 1, 2, 4, 1, 3, 3, 10, 1, 1, 12, 4, 7, 1, 25, 4, 10, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 3, 7, 1, 1, 1, 3, 1, 1, 1, 1, 27]]
[81, [18584021029025603512479467959285372526679, 270, 5, 3, 10, 4, 1, 2, 3, 1, 5, 1, 3, 1, 18, 2, 1, 1, 9, 1, 8, 953, 8, 3, 1, 8, 4]]
[82, [58383423938945065690063367398575102156630, 1, 1, 1, 50, 19, 6, 4, 1, 10, 1, 3, 1, 11, 2, 14, 5, 7, 6, 1, 1, 12, 1, 3, 1, 1, 10, 1, 1, 1, 84]]
[83, [183416935738008287619993279499823507509433, 1, 20, 3, 2, 6, 1, 4, 5, 3, 1, 1, 7, 1, 2, 1, 1, 1, 1, 24, 1, 1, 3, 39, 1, 8, 2, 4, 3, 6, 13, 1, 1, 10]]
[84, [576221297858478037757896202112843514258698, 1, 6, 3, 3, 3, 2, 2, 5, 5, 3, 1, 4, 1, 3, 1, 2, 2, 1, 1, 1, 1, 4, 38, 2, 2, 1, 110, 5, 1, 3, 1, 2, 1, 2, 2]]
[85, [1810252596194170662517162499694380571752661, 5, 2, 3, 1, 7, 1, 2, 1, 2, 1, 4, 1, 2, 2, 4, 1, 5, 2, 4, 1, 3, 1, 50, 1, 3, 48, 1, 7, 2, 1, 1, 1, 6, 1, 4, 5]]
[86, [5687076257345457055936827972911358112818998, 3, 59, 1, 9, 1, 2, 5, 1, 2, 1, 4, 1, 16, 1, 1, 1, 2, 1, 16, 1, 21, 1, 3, 1, 5, 1, 1, 1, 3, 1, 1, 16, 3, 1, 6]]
[87, [17866476990481424293061935394100969629693609, 1, 1, 1, 2, 1, 3, 2, 1, 13, 1, 6, 17, 13, 3, 9, 317, 1, 2, 1, 4, 1, 13, 1, 1, 4, 2, 5, 1, 1, 1, 1, 2]]
[88, [56129192858827520816193436882886842322337670, 25, 11, 2, 1, 2, 12, 9, 7, 1, 95, 99, 3, 1, 4, 228, 1, 5, 6, 5]]
[89, [176335059937217224018860024808747332179340048, 2, 51, 1, 6, 48, 1, 16, 1, 1, 1, 1, 2, 12, 1, 7, 3, 1, 72, 1, 1, 1, 2, 2, 1, 3, 19, 2, 1, 4, 1, 2]]
[90, [553972928869077498297581459358161277972163803, 10, 2, 1, 3, 8, 3, 2, 6, 17, 1, 17, 1, 1, 4, 2, 1, 1673, 1, 1, 1, 2, 2, 2, 1, 79, 3]]
[91, [1740357283622714955277630726239733032237686240, 9, 2, 2, 7, 3, 8, 6, 1, 8, 2, 3, 7, 2, 1, 2, 4, 5, 1, 1, 1, 39, 7, 1, 1, 1, 75, 2]]
[92, [5467493656850609485803318470436356848616987732, 1, 1, 1, 1, 4, 13, 22, 1, 1, 8, 1, 1, 3, 1, 2, 2, 1, 18, 12, 1, 2, 1, 5, 3, 1, 3, 1, 290, 8]]
[93, [17176637905910668669319694724112097956068639929, 16, 2, 6, 1, 14, 1, 12, 1, 6, 3, 6, 4, 1, 1, 1, 2, 1, 19, 1, 30, 1, 1, 1, 98, 1, 1, 1, 3, 2]]
[94, [53961999458580926862279289940854110180185778687, 1, 1, 1, 5, 2, 2, 3, 14, 2, 1, 12, 1, 31, 3, 3, 2, 2, 23, 1, 16, 1, 1, 8, 1, 3, 1, 2, 1, 5]]
[95, [169526621072094240051062813849433542430041251045, 1, 3, 1, 1, 1, 1, 1, 3, 1, 6, 2, 7, 1, 5, 1, 1, 35, 5, 1, 19, 12, 1, 2, 1, 4, 2, 33, 1, 1, 12]]
[96, [532583587347991902715777143640488633081518781241, 1, 2, 1, 1, 1, 2, 5, 1, 1, 3, 1, 2, 13, 1, 1, 1, 1, 6, 1, 52, 1, 2, 9, 3, 1, 1, 1, 16, 1, 1, 3, 1, 5, 10]]
[97, [1673160685434949314609400471476613579121899911697, 1, 11, 1, 1, 1, 1, 1, 40, 1, 1, 1, 4, 1, 2, 1, 14, 3, 3, 2, 1, 4, 1, 1, 9, 1, 36, 1, 2, 5, 1, 1, 4]]
[98, [5256389317637699735335333000834901305449119091054, 1, 2, 1, 19, 8, 1, 8, 1, 3, 2, 1, 2, 1, 2, 9, 1, 5, 12, 4, 1, 1, 2, 7, 1, 8, 1, 3, 3, 1, 3]]
[99, [16513434064698463682619509822252680822186405912861, 3, 3, 2, 9, 1, 1, 1, 6, 3, 1, 21, 1, 1, 24, 1, 68, 1, 11, 2, 36, 3, 1, 1, 2]]
[100, [51878483143196131920862615246303013562686760680405, 1, 3, 1, 1, 1, 6, 2, 2, 18, 1, 5, 11, 2, 1, 2, 1, 1, 3, 2, 16, 3, 1, 4, 3, 1, 3, 2, 4, 2]]
投稿者 時刻: 0 件のコメント:
ラベル:

2024年3月25日月曜日

240325

PARI


シュワルツの定理

「ラマヌジャンの遺した関数」の2.7 に載っている定理を確かめてみる。

a(n, m) = binomial(n^2, n)%n^m;
for(m=1, 6, for(n=1, 50, print1(a(n, m),", ")); print)

出力結果
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 2, 3, 12, 5, 12, 7, 56, 9, 40, 11, 108, 13, 84, 90, 240, 17, 144, 19, 80, 315, 220, 23, 72, 25, 312, 27, 560, 29, 0, 31, 992, 759, 544, 770, 720, 37, 684, 1053, 520, 41, 252, 43, 1408, 1125, 1012, 47, 1872, 49, 1200, 
0, 6, 3, 28, 5, 120, 7, 120, 9, 440, 11, 828, 13, 1064, 315, 496, 17, 2088, 19, 2080, 7812, 3608, 23, 648, 25, 5720, 6588, 10752, 29, 1800, 31, 18400, 29073, 12104, 36295, 12384, 37, 16568, 36036, 54920, 41, 12600, 43, 69168, 74025, 28520, 47, 54864, 49, 36200, 
0, 6, 3, 28, 5, 1200, 7, 3192, 2925, 6440, 11, 828, 13, 14784, 10440, 12784, 17, 25416, 19, 34080, 118944, 205920, 23, 111240, 156275, 286936, 262467, 515648, 29, 244800, 31, 575456, 640002, 169320, 36295, 712224, 37, 1827344, 2230839, 2166920, 41, 2012976, 43, 1091376, 3445650, 1683232, 47, 54864, 823592, 2036200, 
0, 6, 84, 796, 5, 3792, 7, 27768, 22608, 56440, 11, 83772, 13, 168448, 61065, 471536, 17, 445320, 19, 994080, 2258235, 4188272, 23, 6083208, 156275, 743912, 13548492, 3588928, 29, 16444800, 31, 28887008, 13685133, 2841992, 42053795, 49421088, 37, 24763840, 9171162, 45686920, 41, 101587248, 43, 98541872, 15747525, 64367616, 47, 153998928, 58471602, 133286200, 
0, 6, 84, 1820, 6255, 34896, 16814, 126072, 258804, 456440, 805266, 2820924, 2227771, 3933216, 1579815, 7811568, 9939016, 26899272, 12380514, 994080, 47183346, 112414544, 32181738, 22008456, 205234400, 131439048, 157037562, 365006656, 184600370, 210844800, 429437296, 699975648, 796392993, 502631656, 1460144420, 2044804896, 901471478, 1767937536, 2264776137, 1888886920, 2780548865, 1016425872, 735042258, 6860107056, 6843288150, 9332701536, 1834760103, 3211646544, 9380154819, 8883286200, 
投稿者 時刻: 0 件のコメント:
ラベル:

2024年2月25日日曜日

240225

PARI


Expansion of (1/x) * Series_Reversion( x * (1-x)^u * (1-x^s)^t ).

2通りで算出し、一致することを確認してみた。

a(n, s, t, u) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef(serreverse(x*(1-x)^u*(1-x^s)^t)/x, n);
diff(n, s, t, u) = a(n, s, t, u)-b(n, s, t, u);

for(s=1, 2, for(t=-5, 5, for(u=-5, 5, print1([s, t, u],": "); for(n=0, 10, if(diff(n, s, t, u)==0, print1(a(n, s, t, u),", "))); print)))

出力結果
[1, -5, -5]: 1, -10, 145, -2470, 46060, -910252, 18730855, -397089550, 8612835715, -190223180840, 4263421511271, 
[1, -5, -4]: 1, -9, 117, -1785, 29799, -527085, 9706503, -184138713, 3573805950, -70625252863, 1416298046436, 
[1, -5, -3]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 
[1, -5, -2]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 
[1, -5, -1]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 
[1, -5, 0]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 
[1, -5, 1]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[1, -5, 2]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, -5, 3]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, -5, 4]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, -5, 5]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, -4, -5]: 1, -9, 117, -1785, 29799, -527085, 9706503, -184138713, 3573805950, -70625252863, 1416298046436, 
[1, -4, -4]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 
[1, -4, -3]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 
[1, -4, -2]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 
[1, -4, -1]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 
[1, -4, 0]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[1, -4, 1]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, -4, 2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, -4, 3]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, -4, 4]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, -4, 5]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, -3, -5]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 
[1, -3, -4]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 
[1, -3, -3]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 
[1, -3, -2]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 
[1, -3, -1]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[1, -3, 0]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, -3, 1]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, -3, 2]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, -3, 3]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, -3, 4]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, -3, 5]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, -2, -5]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 
[1, -2, -4]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 
[1, -2, -3]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 
[1, -2, -2]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[1, -2, -1]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, -2, 0]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, -2, 1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, -2, 2]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, -2, 3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, -2, 4]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, -2, 5]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, -1, -5]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 
[1, -1, -4]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 
[1, -1, -3]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[1, -1, -2]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, -1, -1]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, -1, 0]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, -1, 1]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, -1, 2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, -1, 3]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, -1, 4]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, -1, 5]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[1, 0, -5]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 
[1, 0, -4]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[1, 0, -3]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, 0, -2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, 0, -1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, 0, 0]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, 0, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, 0, 2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, 0, 3]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, 0, 4]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[1, 0, 5]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 
[1, 1, -5]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[1, 1, -4]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, 1, -3]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, 1, -2]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, 1, -1]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, 1, 0]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, 1, 1]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, 1, 2]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, 1, 3]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[1, 1, 4]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 
[1, 1, 5]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 
[1, 2, -5]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[1, 2, -4]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, 2, -3]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, 2, -2]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, 2, -1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, 2, 0]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, 2, 1]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, 2, 2]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[1, 2, 3]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 
[1, 2, 4]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 
[1, 2, 5]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 
[1, 3, -5]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[1, 3, -4]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, 3, -3]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, 3, -2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, 3, -1]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, 3, 0]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, 3, 1]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[1, 3, 2]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 
[1, 3, 3]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 
[1, 3, 4]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 
[1, 3, 5]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 
[1, 4, -5]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[1, 4, -4]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, 4, -3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, 4, -2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, 4, -1]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, 4, 0]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[1, 4, 1]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 
[1, 4, 2]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 
[1, 4, 3]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 
[1, 4, 4]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 
[1, 4, 5]: 1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 
[1, 5, -5]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[1, 5, -4]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[1, 5, -3]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[1, 5, -2]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[1, 5, -1]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[1, 5, 0]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 
[1, 5, 1]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 
[1, 5, 2]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 
[1, 5, 3]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 
[1, 5, 4]: 1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 
[1, 5, 5]: 1, 10, 155, 2870, 58565, 1270752, 28765650, 671650110, 16057800980, 391139588190, 9672348219898, 
[2, -5, -5]: 1, -5, 30, -185, 1090, -5626, 19720, 45695, -1929840, 26595030, -282947496, 
[2, -5, -4]: 1, -4, 17, -60, 79, 1296, -17360, 142068, -873895, 3711532, -2373973, 
[2, -5, -3]: 1, -3, 7, 5, -192, 1347, -5258, 1557, 160270, -1442000, 6975729, 
[2, -5, -2]: 1, -2, 0, 26, -123, 98, 2224, -14430, 23967, 245364, -2067680, 
[2, -5, -1]: 1, -1, -4, 19, 11, -336, 676, 4699, -26764, -25371, 663520, 
[2, -5, 0]: 1, 0, -5, 0, 60, 0, -935, 0, 16555, 0, -316251, 
[2, -5, 1]: 1, 1, -3, -15, -1, 197, 527, -1711, -13530, -7993, 221780, 
[2, -5, 2]: 1, 2, 2, -10, -72, -222, -34, 3598, 20735, 50630, -109596, 
[2, -5, 3]: 1, 3, 10, 31, 72, -6, -1540, -13065, -78479, -381576, -1474818, 
[2, -5, 4]: 1, 4, 21, 124, 781, 5120, 34474, 236492, 1644222, 11543644, 81623504, 
[2, -5, 5]: 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 
[2, -4, -5]: 1, -5, 31, -205, 1368, -8891, 54102, -286173, 1020910, 2638800, -110587783, 
[2, -4, -4]: 1, -4, 18, -76, 247, -92, -8136, 97092, -798813, 5301608, -28151584, 
[2, -4, -3]: 1, -3, 8, -7, -109, 1008, -5316, 16145, 19220, -665873, 5278504, 
[2, -4, -2]: 1, -2, 1, 18, -100, 196, 871, -8758, 29971, 27512, -858793, 
[2, -4, -1]: 1, -1, -3, 15, -1, -197, 527, 1711, -13530, 7993, 221780, 
[2, -4, 0]: 1, 0, -4, 0, 38, 0, -468, 0, 6545, 0, -98728, 
[2, -4, 1]: 1, 1, -2, -11, -8, 94, 336, -259, -5087, -10712, 41026, 
[2, -4, 2]: 1, 2, 3, -2, -39, -176, -442, -26, 6222, 36062, 113240, 
[2, -4, 3]: 1, 3, 11, 43, 170, 657, 2392, 7675, 17603, -11898, -529678, 
[2, -4, 4]: 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 
[2, -4, 5]: 1, 5, 36, 305, 2833, 27916, 286632, 3033513, 32858595, 362515725, 4059475368, 
[2, -3, -5]: 1, -5, 32, -225, 1651, -12336, 92460, -687161, 5006180, -35235655, 234112824, 
[2, -3, -4]: 1, -4, 19, -92, 420, -1624, 3545, 20372, -386621, 3865104, -31129539, 
[2, -3, -3]: 1, -3, 9, -19, -21, 561, -4093, 20493, -69030, 43819, 1571700, 
[2, -3, -2]: 1, -2, 2, 10, -72, 222, -34, -3598, 20735, -50630, -109596, 
[2, -3, -1]: 1, -1, -2, 11, -8, -94, 336, 259, -5087, 10712, 41026, 
[2, -3, 0]: 1, 0, -3, 0, 21, 0, -190, 0, 1950, 0, -21576, 
[2, -3, 1]: 1, 1, -1, -7, -10, 27, 152, 169, -949, -4286, -2646, 
[2, -3, 2]: 1, 2, 4, 6, -1, -58, -304, -1090, -2876, -4216, 9244, 
[2, -3, 3]: 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 8414640, 
[2, -3, 4]: 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 
[2, -3, 5]: 1, 5, 37, 325, 3141, 32261, 345605, 3818501, 43197445, 497868805, 5825331205, 
[2, -2, -5]: 1, -5, 33, -245, 1939, -15961, 134843, -1159829, 10103730, -88820835, 785856036, 
[2, -2, -4]: 1, -4, 20, -108, 598, -3300, 17732, -90140, 411281, -1451980, 965192, 
[2, -2, -3]: 1, -3, 10, -31, 72, 6, -1540, 13065, -78479, 381576, -1474818, 
[2, -2, -2]: 1, -2, 3, 2, -39, 176, -442, 26, 6222, -36062, 113240, 
[2, -2, -1]: 1, -1, -1, 7, -10, -27, 152, -169, -949, 4286, -2646, 
[2, -2, 0]: 1, 0, -2, 0, 9, 0, -52, 0, 340, 0, -2394, 
[2, -2, 1]: 1, 1, 0, -3, -7, -4, 24, 85, 99, -215, -1196, 
[2, -2, 2]: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 
[2, -2, 3]: 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 
[2, -2, 4]: 1, 4, 24, 172, 1360, 11444, 100520, 911068, 8457504, 80006116, 768464312, 
[2, -2, 5]: 1, 5, 38, 345, 3454, 36786, 408848, 4687969, 55048310, 658645110, 8001060132, 
[2, -1, -5]: 1, -5, 34, -265, 2232, -19766, 181300, -1706737, 16392049, -159959240, 1581278838, 
[2, -1, -4]: 1, -4, 21, -124, 781, -5120, 34474, -236492, 1644222, -11543644, 81623504, 
[2, -1, -3]: 1, -3, 11, -43, 170, -657, 2392, -7675, 17603, 11898, -529678, 
[2, -1, -2]: 1, -2, 4, -6, -1, 58, -304, 1090, -2876, 4216, 9244, 
[2, -1, -1]: 1, -1, 0, 3, -7, 4, 24, -85, 99, 215, -1196, 
[2, -1, 0]: 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42, 
[2, -1, 1]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
[2, -1, 2]: 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 
[2, -1, 3]: 1, 3, 14, 79, 494, 3294, 22952, 165127, 1217270, 9146746, 69799476, 
[2, -1, 4]: 1, 4, 25, 188, 1563, 13840, 127972, 1221260, 11938471, 118936100, 1203155633, 
[2, -1, 5]: 1, 5, 39, 365, 3772, 41491, 476410, 5644477, 68493324, 846937140, 10633195119, 
[2, 0, -5]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 
[2, 0, -4]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 
[2, 0, -3]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 
[2, 0, -2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 
[2, 0, -1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 
[2, 0, 0]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
[2, 0, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 
[2, 0, 2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 
[2, 0, 3]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 
[2, 0, 4]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 
[2, 0, 5]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 
[2, 1, -5]: 1, -5, 36, -305, 2833, -27916, 286632, -3033513, 32858595, -362515725, 4059475368, 
[2, 1, -4]: 1, -4, 23, -156, 1162, -9192, 75819, -644908, 5616182, -49826712, 448771622, 
[2, 1, -3]: 1, -3, 13, -67, 381, -2307, 14589, -95235, 636925, -4341763, 30056445, 
[2, 1, -2]: 1, -2, 6, -22, 90, -394, 1806, -8558, 41586, -206098, 1037718, 
[2, 1, -1]: 1, -1, 2, -5, 14, -42, 132, -429, 1430, -4862, 16796, 
[2, 1, 0]: 1, 0, 1, 0, 3, 0, 12, 0, 55, 0, 273, 
[2, 1, 1]: 1, 1, 3, 9, 32, 119, 466, 1881, 7788, 32868, 140907, 
[2, 1, 2]: 1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 
[2, 1, 3]: 1, 3, 16, 103, 735, 5592, 44452, 364815, 3067558, 26290517, 228819168, 
[2, 1, 4]: 1, 4, 27, 220, 1984, 19064, 191325, 1981932, 21031965, 227463808, 2498039219, 
[2, 1, 5]: 1, 5, 41, 405, 4423, 51441, 624687, 7828853, 100497980, 1314688375, 17462945394, 
[2, 2, -5]: 1, -5, 37, -325, 3141, -32261, 345605, -3818501, 43197445, -497868805, 5825331205, 
[2, 2, -4]: 1, -4, 24, -172, 1360, -11444, 100520, -911068, 8457504, -80006116, 768464312, 
[2, 2, -3]: 1, -3, 14, -79, 494, -3294, 22952, -165127, 1217270, -9146746, 69799476, 
[2, 2, -2]: 1, -2, 7, -30, 143, -728, 3876, -21318, 120175, -690690, 4032015, 
[2, 2, -1]: 1, -1, 3, -9, 32, -119, 466, -1881, 7788, -32868, 140907, 
[2, 2, 0]: 1, 0, 2, 0, 11, 0, 80, 0, 665, 0, 5980, 
[2, 2, 1]: 1, 1, 4, 13, 55, 232, 1052, 4869, 23206, 112519, 554560, 
[2, 2, 2]: 1, 2, 9, 46, 264, 1612, 10291, 67830, 458109, 3153744, 22049065, 
[2, 2, 3]: 1, 3, 17, 115, 863, 6903, 57687, 497683, 4398980, 39630305, 362562226, 
[2, 2, 4]: 1, 4, 28, 236, 2202, 21892, 227324, 2436508, 26752627, 299369532, 3401762928, 
[2, 2, 5]: 1, 5, 42, 425, 4756, 56686, 705500, 9061841, 119225535, 1598582580, 21763800786, 
[2, 3, -5]: 1, -5, 38, -345, 3454, -36786, 408848, -4687969, 55048310, -658645110, 8001060132, 
[2, 3, -4]: 1, -4, 25, -188, 1563, -13840, 127972, -1221260, 11938471, -118936100, 1203155633, 
[2, 3, -3]: 1, -3, 15, -91, 612, -4389, 32890, -254475, 2017356, -16301164, 133767543, 
[2, 3, -2]: 1, -2, 8, -38, 201, -1134, 6688, -40734, 254237, -1617572, 10452416, 
[2, 3, -1]: 1, -1, 4, -13, 55, -232, 1052, -4869, 23206, -112519, 554560, 
[2, 3, 0]: 1, 0, 3, 0, 24, 0, 253, 0, 3045, 0, 39627, 
[2, 3, 1]: 1, 1, 5, 17, 83, 381, 1939, 9905, 52544, 282315, 1545130, 
[2, 3, 2]: 1, 2, 10, 54, 332, 2162, 14734, 103630, 746857, 5486206, 40926152, 
[2, 3, 3]: 1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 
[2, 3, 4]: 1, 4, 29, 252, 2425, 24864, 266270, 2943308, 33329204, 384641836, 4507577438, 
[2, 3, 5]: 1, 5, 43, 445, 5094, 62111, 790828, 10392109, 139883065, 1918862510, 26727571728, 
[2, 4, -5]: 1, -5, 39, -365, 3772, -41491, 476410, -5644477, 68493324, -846937140, 10633195119, 
[2, 4, -4]: 1, -4, 26, -204, 1771, -16380, 158224, -1577532, 16112057, -167710664, 1772645420, 
[2, 4, -3]: 1, -3, 16, -103, 735, -5592, 44452, -364815, 3067558, -26290517, 228819168, 
[2, 4, -2]: 1, -2, 9, -46, 264, -1612, 10291, -67830, 458109, -3153744, 22049065, 
[2, 4, -1]: 1, -1, 5, -17, 83, -381, 1939, -9905, 52544, -282315, 1545130, 
[2, 4, 0]: 1, 0, 4, 0, 42, 0, 580, 0, 9139, 0, 155664, 
[2, 4, 1]: 1, 1, 6, 21, 116, 566, 3176, 17501, 101391, 590756, 3519782, 
[2, 4, 2]: 1, 2, 11, 62, 405, 2784, 20066, 149158, 1136276, 8821958, 69556554, 
[2, 4, 3]: 1, 3, 19, 139, 1134, 9849, 89372, 837147, 8033545, 78573186, 780388752, 
[2, 4, 4]: 1, 4, 30, 268, 2653, 27980, 308212, 3504380, 40817586, 484534720, 5840593274, 
[2, 4, 5]: 1, 5, 44, 465, 5437, 67716, 880720, 11822217, 162556349, 2277870665, 32411053200, 
[2, 5, -5]: 1, -5, 40, -385, 4095, -46376, 548340, -6690585, 83615350, -1064887395, 13770292256, 
[2, 5, -4]: 1, -4, 27, -220, 1984, -19064, 191325, -1981932, 21031965, -227463808, 2498039219, 
[2, 5, -3]: 1, -3, 17, -115, 863, -6903, 57687, -497683, 4398980, -39630305, 362562226, 
[2, 5, -2]: 1, -2, 10, -54, 332, -2162, 14734, -103630, 746857, -5486206, 40926152, 
[2, 5, -1]: 1, -1, 6, -21, 116, -566, 3176, -17501, 101391, -590756, 3519782, 
[2, 5, 0]: 1, 0, 5, 0, 65, 0, 1110, 0, 21620, 0, 455126, 
[2, 5, 1]: 1, 1, 7, 25, 154, 787, 4812, 28169, 176065, 1096342, 7020112, 
[2, 5, 2]: 1, 2, 12, 70, 483, 3478, 26336, 205438, 1642890, 13388000, 110796476, 
[2, 5, 3]: 1, 3, 20, 151, 1277, 11484, 107920, 1046815, 10402541, 105357279, 1083566504, 
[2, 5, 4]: 1, 4, 31, 284, 2886, 31240, 353199, 4121772, 49274392, 600342184, 7427285236, 
[2, 5, 5]: 1, 5, 45, 485, 5785, 73501, 975225, 13354725, 187331895, 2677999545, 38873136507, 
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