A392892 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + (exp(x^3) - 1)^2/x^2) ).

1, 0, 0, 0, 24, 0, 0, 5040, 161280, 0, 2116800, 439084800, 10538035200, 1556755200, 1322204083200, 137305808640000, 2930992269004800, 5038905231360000, 1550975030212608000, 117877143423668428800, 2382115178755952640000, 18775921248103219200000

Offset

0,5

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))^3) - 1)^2/(x*A(x))^2.

a(n) = (n!)^2 * Sum_{k=0..floor(n/3)} (2*(n-3*k))!/((n-3*k)! * (3*k+1)!) * Stirling2(n-2*k,2*(n-3*k))/(n-2*k)!.

Mathematica

Table[(n!)^2*Sum[(2*(n-3*k))!/((n-3*k)!*(3*k+1)!)*Abs[StirlingS2[n-2*k, 2*(n-3*k)]/(n-2*k)!], {k, 0, Floor[n/3]}], {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^3)-1)^2/x^2))/x))
(Magma) [ Factorial(n)^2 * &+[Factorial(2*(n - 3*k)) / (Factorial(n - 3*k) * Factorial(3*k + 1)) * Abs(StirlingSecond(n - 2*k, 2*(n - 3*k)) / Factorial(n - 2*k)): k in [0..Floor(n/3)]] : n in [0..23] ]; // Vincenzo Librandi, Jan 26 2026

Crossrefs

Cf. A392849, A392891.

Sequence in context: A392938 A376347 A376346 * A392888 A392795 A392792

Adjacent sequences: A392889 A392890 A392891 * A392893 A392894 A392895

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved