A381890 Expansion of Product_{k>=1} (1 + k*x)^((1/12) * (3/4)^k).

1, 1, -3, 21, -225, 3207, -56821, 1202099, -29558466, 828401462, -26068940938, 910286433318, -34930741605414, 1461245816594058, -66187658069563710, 3227353484661602866, -168557942284281821933, 9388117645333487820387, -555463036269652132509113

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• sign

Crossrefs

Cf. A384343, A384344, A384345.

Cf. A050352, A090353.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 3^(j-1)*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/4) * A(x/(1+x))^(3/4).

G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A050352(k) * x^k/k).

G.f.: 1/B(-x), where B(x) is the g.f. of A090353.