A384324 Expansion of Product_{k>=1} 1/(1 - k*x)^((2/3)^k).

1, 6, 33, 200, 1428, 12408, 132604, 1730160, 27043866, 495026316, 10388326986, 245555445888, 6446710871724, 185904786328920, 5836500883321164, 198054400887909264, 7220679972923312487, 281402128806812402490, 11671796413017231008663

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• nonn

Crossrefs

Cf. A084785, A384305, A384325, A384326.

Cf. A004123, A090351, A090352.

Programs

Mathematica

terms = 20; A[] = 1; Do[A[x] = -2*A[x] + 3*A[x/(1-x)]^(2/3) / (1-x)^2 + 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(3*sum(k=1, N, sum(j=0, k, 2^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = A(x/(1-x))^(2/3) / (1-x)^2.

G.f.: exp(3 * Sum_{k>=1} A004123(k+1) * x^k/k).

G.f.: B(x)^6, where B(x) is the g.f. of A090351.

a(n) ~ (n-1)! / log(3/2)^(n+1). - Vaclav Kotesovec, May 27 2025