A392830 E.g.f. A(x) satisfies A(x) = 1/(1 - x*A(x)*log(1-x)^2).

1, 0, 0, 6, 24, 110, 2040, 23996, 256704, 4589784, 85865760, 1525452192, 33101418240, 805692925440, 19851600926976, 536507047398240, 15942291085808640, 494403405864883200, 16257895376916506112, 573326016462139601664, 21243600967453427128320, 825607221009339857034240

Offset

0,4

Links

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

Formula

E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * log(1-x)^2.

E.g.f.: 2/(1 + sqrt(1-4*x*log(1-x)^2)).

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

Mathematica

Table[n!* Sum[(2*k)!*Binomial[2*k+1, k]/(2*k+1)*Abs[StirlingS1[n-k, 2*k]]/(n-k)!, {k, 0, Floor[n/3]}], {n, 0, 21}] (* Vincenzo Librandi, Feb 07 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (2*k)!*binomial(2*k+1, k)/(2*k+1)*abs(stirling(n-k, 2*k, 1))/(n-k)!);
(Magma) [Factorial(n) * &+[Factorial(2*k)* Binomial(2*k+1, k)/(2*k+1) * Abs(StirlingFirst(n - k, 2*k)) / Factorial(n - k): k in [0..Floor(n/3)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 07 2026

Crossrefs

Cf. A000108, A392760.

Sequence in context: A392823 A392915 A392916 * A392829 A392760 A392856

Adjacent sequences: A392827 A392828 A392829 * A392831 A392832 A392833

Keyword

nonn

Author

Seiichi Manyama, Jan 24 2026

Status

approved