A392825 E.g.f. A(x) satisfies A(x) = 1/(1 - x*(exp(x*A(x)) - 1)^2).

1, 0, 0, 6, 24, 70, 2340, 35714, 350448, 7078446, 181671900, 3588919642, 82318065192, 2517758565062, 74690172967956, 2233128307655730, 78192431795237856, 2939077530723101278, 111643916383932333516, 4566683724663387332618, 202009025773688741277720

Offset

0,4

Links

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

Formula

E.g.f.: (1/x) * Series_Reversion( x/(1 + x*(exp(x) - 1)^2) ).

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

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

Mathematica

Table[n!/(n+1)*Sum[(2*k)!*Binomial[n+1, k]*Abs[StirlingS2[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)!*binomial(n+1, k)*stirling(n-k, 2*k, 2)/(n-k)!)/(n+1);
(Magma) [Factorial(n)/(n+1)* &+[Factorial(2*k)*Binomial(n+1, k)*Abs(StirlingSecond(n-k, 2*k))/Factorial(n-k) : k in [0..Floor(n/3)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 25 2026

Crossrefs

Cf. A371119, A392766.

Sequence in context: A052749 A392820 A392826 * A392766 A392855 A379825

Adjacent sequences: A392822 A392823 A392824 * A392826 A392827 A392828

Keyword

nonn

Author

Seiichi Manyama, Jan 24 2026

Status

approved