A384305 Expansion of Product_{k>=1} 1/(1 - k*x)^((5/6)^k).

1, 30, 615, 11260, 205695, 4013406, 88035585, 2255192280, 68859250020, 2506898720040, 107238427737876, 5281094776037040, 293625956135692020, 18139856902224931080, 1229886945212115522060, 90641666662687182976896, 7206758883035555464430370, 614391718014749017022916060

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• nonn

Crossrefs

Cf. A084785, A384324, A384325, A384326.

Cf. A090358, A090362, A094418.

Programs

Mathematica

terms = 20; A[] = 1; Do[A[x] = -5*A[x] + 6*A[x/(1-x)]^(5/6) / (1-x)^5 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 31 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(exp(6*sum(k=1, N, sum(j=0, k, 5^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = A(x/(1-x))^(5/6) / (1-x)^5.

G.f.: exp(6 * Sum_{k>=1} A094418(k) * x^k/k).

G.f.: B(x)^30, where B(x) is the g.f. of A090358.

a(n) ~ (n-1)! / log(6/5)^(n+1). - Vaclav Kotesovec, May 31 2025