201115

PARI


A338864, A338865, A338870 とA338871

それぞれのT(n;u) に関する関係式を確認してみた。

T(n) = if(n==0, 1, u*(n-1)!*sum(k=1, n, k*numdiv(k)*T(n-k)/(n-k)!));
print("A338864");
for(n=0, 10, print(n, " ", Vecrev(T(n))))

T(n) = if(n==0, 1, u*(n-1)!*sum(k=1, n, k*sigma(k)*T(n-k)/(n-k)!));
print("A338865");
for(n=0, 10, print(n, " ", Vecrev(T(n))))

T(n) = if(n==0, 1, u*sum(k=1, n, binomial(n-1, k-1)*numdiv(k)*T(n-k)));
print("A338870");
for(n=0, 10, print(n, " ", Vecrev(T(n))))

T(n) = if(n==0, 1, u*sum(k=1, n, binomial(n-1, k-1)*sigma(k)*T(n-k)));
print("A338871");
for(n=0, 10, print(n, " ", Vecrev(T(n))))

出力結果

A338864

0 [1]

1 [0, 1]

2 [0, 4, 1]

3 [0, 12, 12, 1]

4 [0, 72, 96, 24, 1]

5 [0, 240, 840, 360, 40, 1]

6 [0, 2880, 7200, 4920, 960, 60, 1]

7 [0, 10080, 70560, 65520, 19320, 2100, 84, 1]

8 [0, 161280, 745920, 887040, 362880, 58800, 4032, 112, 1]

9 [0, 1088640, 7983360, 12640320, 6652800, 1481760, 150192, 7056, 144, 1]

10 [0, 14515200, 105235200, 185068800, 123076800, 35441280, 4898880, 337680, 11520, 180, 1]

A338865

0 [1]

1 [0, 1]

2 [0, 6, 1]

3 [0, 24, 18, 1]

4 [0, 168, 204, 36, 1]

5 [0, 720, 2280, 780, 60, 1]

6 [0, 8640, 25200, 14400, 2100, 90, 1]

7 [0, 40320, 292320, 252000, 58800, 4620, 126, 1]

8 [0, 604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1]

9 [0, 4717440, 46811520, 76265280, 35743680, 6335280, 474768, 15624, 216, 1]

10 [0, 65318400, 696729600, 1368057600, 844603200, 203001120, 21576240, 1078560, 25560, 270, 1]

A338870

0 [1]

1 [0, 1]

2 [0, 2, 1]

3 [0, 2, 6, 1]

4 [0, 3, 20, 12, 1]

5 [0, 2, 55, 80, 20, 1]

6 [0, 4, 142, 405, 220, 30, 1]

7 [0, 2, 322, 1792, 1785, 490, 42, 1]

8 [0, 4, 779, 7224, 12152, 5810, 952, 56, 1]

9 [0, 3, 1608, 27323, 73920, 56532, 15498, 1680, 72, 1]

10 [0, 4, 3894, 99690, 414815, 482160, 204204, 35910, 2760, 90, 1]

A338871

0 [1]

1 [0, 1]

2 [0, 3, 1]

3 [0, 4, 9, 1]

4 [0, 7, 43, 18, 1]

5 [0, 6, 155, 175, 30, 1]

6 [0, 12, 511, 1230, 485, 45, 1]

7 [0, 8, 1442, 7231, 5600, 1085, 63, 1]

8 [0, 15, 4131, 37870, 52381, 18550, 2114, 84, 1]

9 [0, 13, 10323, 181063, 426006, 253281, 50022, 3738, 108, 1]

10 [0, 18, 28171, 818760, 3128245, 2956065, 937587, 116760, 6150, 135, 1]