A384334 Expansion of Product_{k>=1} (1 + k*x)^((4/5)^k).

1, 20, 110, 340, -1995, 53904, -1534600, 49159600, -1758057650, 69662897000, -3037327435860, 144787947993000, -7502235351828450, 420296374337607600, -25335189019626256200, 1636008982452733508400, -112721505676611504401025, 8256863266451569604835900

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• sign

Crossrefs

Cf. A116603, A384332, A384333.

Cf. A094417, A384326, A384345.

Programs

Mathematica

terms = 20; A[] = 1; Do[A[x] = -4*A[x] + 5*A[x/(1+x)]^(4/5) * (1+x)^4 + 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(5*sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 4^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

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

G.f.: exp(5 * Sum_{k>=1} (-1)^(k-1) * A094417(k) * x^k/k).

G.f.: 1/B(-x), where B(x) is the g.f. of A384326.

G.f.: B(x)^20, where B(x) is the g.f. of A384345.

a(n) ~ (-1)^(n+1) * (n-1)! / log(5/4)^(n+1). - Vaclav Kotesovec, May 27 2025