A384326 Expansion of Product_{k>=1} 1/(1 - k*x)^((4/5)^k).

1, 20, 290, 3940, 55695, 872904, 15862460, 343510120, 8931896095, 276115329860, 9954870557826, 410042908659060, 18954497571869745, 969420292296268320, 54253252462944958560, 3293672518482920204544, 215400856153695252763320, 15088195059520554250863840

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• nonn

Crossrefs

Cf. A084785, A384305, A384324, A384325.

Cf. A090356, A094417.

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, sum(j=0, k, 4^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

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

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

G.f.: B(x)^20, where B(x) is the g.f. of A090356.

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