A384325 Expansion of Product_{k>=1} 1/(1 - k*x)^((3/4)^k).

1, 12, 114, 1084, 11319, 136920, 1981228, 34705656, 731268315, 18203860748, 524073230394, 17111173850652, 623571696107069, 25046605210733184, 1097919954149781264, 52109508350206511840, 2660615337817983390318, 145353541761618312219336

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• nonn

Crossrefs

Cf. A084785, A384305, A384324, A384326.

Cf. A032033, A090353.

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, sum(j=0, k, 3^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

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

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

G.f.: B(x)^12, where B(x) is the g.f. of A090353.

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