A392824 Expansion of e.g.f. 1/(1 + x*log(1-x)^3).

1, 0, 0, 0, 24, 180, 1260, 9450, 118272, 1797768, 27953040, 436649400, 7410724992, 141112013328, 2956403163168, 65916451823040, 1548854017058304, 38540253128567040, 1019365138785173760, 28554827581522460160, 841931111707235942400, 26025266490517000788480

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (3*k)! * |Stirling1(n-k,3*k)|/(n-k)!.

a(n) ~ n! * (1-r) / ((1 - r + 3*r^(4/3)) * r^n), where r = 0.67938820663137884521135112994538... is the root of the equation r*log(1-r)^3 = -1. - Vaclav Kotesovec, Jan 25 2026

Mathematica

Table[n!*Sum[(3*k)!*Abs[StirlingS1[n-k, 3*k]/(n-k)!], {k, 0, Floor[n/4]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 25 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\4, (3*k)!*abs(stirling(n-k, 3*k, 1))/(n-k)!);
(Magma) [Factorial(n)* &+[Factorial(3*k)*Abs(StirlingFirst(n-k, 3*k))/Factorial(n-k) : k in [0..Floor(n/4)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 25 2026

Crossrefs

Column k=3 of A392822.

Sequence in context: A073993 A214310 A052758 * A392918 A392917 A392854

Adjacent sequences: A392821 A392822 A392823 * A392825 A392826 A392827

Keyword

nonn,easy

Author

Seiichi Manyama, Jan 24 2026

Status

approved