A384345 Expansion of Product_{k>=1} (1 + k*x)^((1/20) * (4/5)^k).

1, 1, -4, 36, -494, 9026, -205284, 5581276, -176518189, 6366839811, -257967985400, 11601382088720, -573484266103260, 30909105184132900, -1804012437852543160, 113356419526025564808, -7629831521445348113927, 547688013439312943707673, -41765446604358525581076812

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• sign

Crossrefs

Cf. A381890, A384343, A384344.

Cf. A050353, A090356.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 4^(j-1)*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/5) * A(x/(1+x))^(4/5).

G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A050353(k) * x^k/k).

G.f.: 1/B(-x), where B(x) is the g.f. of A090356.