A392823 Expansion of e.g.f. 1/(1 - x*log(1-x)^2).

1, 0, 0, 6, 24, 110, 1320, 13916, 142464, 1868184, 27699120, 429736032, 7337721600, 137702448000, 2762574604608, 59222612250720, 1358702356193280, 33117541456204800, 853766305419430656, 23241033325798924032, 666138680618587407360, 20043625289732615531520

Offset

0,4

Links

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

Formula

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

a(n) ~ n! * (1 - r^2) / ((1 - r^2 + 2*r^3) * r^(2*n)), where r = 0.83540815864975859903313795180514716830912765826... is the root of the equation 1-r^2 = exp(-1/r). - Vaclav Kotesovec, Jan 25 2026

Mathematica

Table[n!*Sum[(2*k)!*Abs[StirlingS1[n-k, 2*k]/(n-k)!], {k, 0, Floor[n/3]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 25 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (2*k)!*abs(stirling(n-k, 2*k, 1))/(n-k)!);
(Magma) [Factorial(n)* &+[Factorial(2*k)*Abs(StirlingFirst(n-k, 2*k))/Factorial(n-k) : k in [0..Floor(n/3)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 25 2026

Crossrefs

Column k=2 of A392822.

Sequence in context: A122739 A038380 A052745 * A392915 A392916 A392830

Adjacent sequences: A392820 A392821 A392822 * A392824 A392825 A392826

Keyword

nonn,easy

Author

Seiichi Manyama, Jan 24 2026

Status

approved