A392891 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + (exp(x^2) - 1)^2/x) ).

1, 0, 0, 6, 0, 120, 2160, 2940, 322560, 4445280, 39312000, 2198861280, 31135104000, 769192744320, 32350774135680, 597288505413600, 24495937449984000, 919567111913049600, 24828597683118028800, 1240480507369945121280, 47003467274263019520000, 1804660990139653802956800

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 + (exp((x*A(x))^2) - 1)^2/(x*A(x)).

a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} (2*(n-2*k))!/((n-2*k)! * (2*k+1)!) * Stirling2(n-k,2*(n-2*k))/(n-k)!.

Mathematica

Table[(n!)^2*Sum[(2*(n-2*k))!/((n-2*k)!*(2*k+1)!)*Abs[StirlingS2[n-k, 2*(n-2*k)]/(n-k)!], {k, 0, Floor[n/2]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 26 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+(exp(x^2)-1)^2/x))/x))
(Magma) [ Factorial(n)^2 * &+[Factorial(2*(n - 2*k)) / (Factorial(n - 2*k) * Factorial(2*k + 1)) * Abs(StirlingSecond(n - k, 2*(n - 2*k)) / Factorial(n - k)): k in [0..Floor(n/2)]]: n in [0..25] ]; // Vincenzo Librandi, Jan 26 2026

Crossrefs

Cf. A392849, A392892.

Sequence in context: A167028 A246137 A052679 * A392887 A392794 A392791

Adjacent sequences: A392888 A392889 A392890 * A392892 A392893 A392894

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved