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]