A392851 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*(exp(x) - 1)^3) ).

1, 0, 0, 0, 24, 180, 900, 3780, 175728, 4951044, 86365500, 1153213380, 23556955752, 859608051588, 28980467230260, 782915612415300, 20550183180508896, 708162598061362692, 30250472697427079340, 1248019079468428688580, 47622250712947210524120, 1899094232159722382052996

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) * (exp(x*A(x)) - 1)^3.

E.g.f. A(x) satisfies A(x) = 1/(1 - x*(exp(x*A(x)) - 1)^3).

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/4)} (3*k)! * binomial(n+1,k) * Stirling2(n-k,3*k)/(n-k)!.

Mathematica

Table[n!/(n+1)*Sum[(3*k)!*Binomial[n+1, k]*Abs[StirlingS2[n-k, 3*k]/(n-k)!], {k, 0, Floor[n/4]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 25 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\4, (3*k)!*binomial(n+1, k)*stirling(n-k, 3*k, 2)/(n-k)!)/(n+1);
(Magma) [Factorial(n)/(n+1)* &+[Factorial(3*k)*Binomial(n+1, k)*Abs(StirlingSecond(n-k, 3*k))/Factorial(n-k) : k in [0..Floor(n/3)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 25 2026

Crossrefs

Cf. A371119, A392825.

Cf. A392828, A392850.

Sequence in context: A165187 A052761 A392821 * A371158 A371198 A073993

Adjacent sequences: A392848 A392849 A392850 * A392852 A392853 A392854

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved