250727

PARI


A386558

f_k(x) = x*Sum_{j>=0} binomial((k+1)*j+k-1, j)/(j+1) * x^j

がf_k(x) * (1 - f_k(x))^k = x を満たすことを確認してみた。

N=10;
K=10;

a(n, k) = binomial((k+1)*n+k-1, n)/(n+1);
b(n, k) = my(x='x+O('x^(n+1))); x*sum(j=0, n, a(j, k)*x^j);
for(k=0, K, f=b(N, k); if(f*(1-f)^k==x, print([k, Vec(f)])));

出力結果

[0, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

[1, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]]

[2, [1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690]]

[3, [1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164]]

[4, [1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664]]

[5, [1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395]]

[6, [1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512]]

[7, [1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245]]

[8, [1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024]]

[9, [1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604]]

[10, [1, 10, 155, 2870, 58565, 1270752, 28765650, 671650110, 16057800980, 391139588190]]