A375688 Expansion of e.g.f. 1 / (1 + 3 * x * log(1 - x))^(1/3).

1, 0, 2, 3, 56, 270, 4824, 44520, 866816, 12195792, 267873120, 5073187680, 126754229568, 2999710359360, 85061489235072, 2400155295632640, 76724104598031360, 2502434971473937920, 89428428468644493312, 3300036525511418327040

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (Product_{j=0..k-1} (3*j+1)) * |Stirling1(n-k,k)|/(n-k)!.

a(n) ~ sqrt(2*Pi) * 3^n * r^(n + 1/3) * n^(n - 1/6) / (Gamma(1/3) * (1/(3*r-1) + log(3*r/(3*r-1)))^(1/3) * exp(n)), where r = 0.67741090606491714108599100770859788... is the root of the equation 3*r*(1 - exp(-r)) = 1. - Vaclav Kotesovec, Jan 23 2026

Mathematica

With[{nn=20}, CoefficientList[Series[1/CubeRoot[1+3x Log[1-x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 21 2025 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*x*log(1-x))^(1/3)))
(PARI) a(n) = n!*sum(k=0, n\2, prod(j=0, k-1, 3*j+1)*abs(stirling(n-k, k, 1))/(n-k)!);

Crossrefs

Cf. A367879, A375689.

Cf. A347015.

Sequence in context: A041709 A361095 A362835 * A371140 A371121 A371227

Adjacent sequences: A375685 A375686 A375687 * A375689 A375690 A375691

Keyword

nonn

Author

Seiichi Manyama, Aug 24 2024

Status

approved