2024年11月24日日曜日

241124

PARI


A135539

Sum_{j>=1} x^(k*j)/(1 - x^j) = Sum_{j>=k} x^j/(1 - x^j) を確認してみた。

N=30;

a(k) = sum(j=1, N, x^(k*j)/(1-x^j) + x*O(x^N));
b(k) = sum(j=k, N, x^j    /(1-x^j) + x*O(x^N));
for(k=1, 10, if(a(k)==b(k), print(k, " ", Vec(a(k)))));

出力結果
1 [1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8]
2 [1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7]
3 [1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 4, 3, 2, 1, 6, 2, 2, 3, 4, 1, 6]
4 [1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5]
5 [1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5]
6 [1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4]
7 [1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3]
8 [1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3]
9 [1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3]
10 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3]
投稿者 時刻: 0 件のコメント:
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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:
ラベル:

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]]
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[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]]
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[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]]
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[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]]
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[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]]
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[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]]
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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, 
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