A384344 Expansion of Product_{k>=1} (1 + k*x)^((1/6) * (2/3)^k).

1, 1, -2, 10, -77, 787, -9972, 150552, -2637729, 52615903, -1177590290, 29228602546, -796945212035, 23681656958269, -761803800466856, 26376749702235900, -978091742247376932, 38674335439691203644, -1624351949069462807480, 72221688529265896447384

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• sign

Crossrefs

Cf. A381890, A384343, A384345.

Cf. A050351, A090351.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 2^(j-1)*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/3) * A(x/(1+x))^(2/3).

G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A050351(k) * x^k/k).

G.f.: 1/B(-x), where B(x) is the g.f. of A090351.