A392827 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^2*(exp(x) - 1)^2) ).

1, 0, 0, 0, 24, 120, 420, 1260, 164752, 3274992, 39334860, 365960100, 13448104824, 506507920776, 12698588406868, 241144625439900, 6724831079454240, 294100019566256352, 11533851799031413404, 361096269237637587540, 11719836491075946081160, 515556895377062422147320

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

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/4)} (2*k)! * binomial(n+1,k) * Stirling2(n-2*k,2*k)/(n-2*k)!.

Mathematica

Table[(n!/(n+1))*Sum[(2*k)!*Binomial[n+1, k]*Abs[StirlingS2[n-2*k, 2*k]/(n-2*k)!], {k, 0, Floor[n/4]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 26 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\4, (2*k)!*binomial(n+1, k)*stirling(n-2*k, 2*k, 2)/(n-2*k)!)/(n+1);
(Magma) [(Factorial(n)/(n+1))* &+[Factorial(2*k)*Binomial(n+1, k)*Abs(StirlingSecond(n-2*k, 2*k))/Factorial(n-2*k): k in [0..Floor(n/4)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 26 2026

Crossrefs

Cf. A371139, A392828.

Sequence in context: A292969 A292979 A052760 * A392767 A179720 A235702

Adjacent sequences: A392824 A392825 A392826 * A392828 A392829 A392830

Keyword

nonn

Author

Seiichi Manyama, Jan 24 2026

Status

approved