A384333 Expansion of Product_{k>=1} (1 + k*x)^((3/4)^k).

1, 12, 30, 76, -819, 15120, -320568, 7719984, -208986462, 6300545128, -209806494828, 7660698340008, -304718887446110, 13127557400200944, -609336227455254936, 30330991088734345200, -1612305658103085757467, 91179423240593288760396, -5467060640706775435713298

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• sign

Crossrefs

Cf. A116603, A384332, A384334.

Cf. A032033, A381890, A384325.

Programs

Mathematica

terms = 20; A[] = 1; Do[A[x] = -3*A[x] + 4*A[x/(1+x)]^(3/4) * (1+x)^3 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 27 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(exp(4*sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 3^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

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

G.f.: exp(4 * Sum_{k>=1} (-1)^(k-1) * A032033(k) * x^k/k).

G.f.: 1/B(-x), where B(x) is the g.f. of A384325.

G.f.: B(x)^12, where B(x) is the g.f. of A381890.

a(n) ~ (-1)^(n+1) * (n-1)! / log(4/3)^(n+1). - Vaclav Kotesovec, May 27 2025