A392769 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - (exp(x)-1)^3) ).

1, 0, 0, 6, 36, 150, 3420, 69846, 963396, 18479670, 523238220, 13553670966, 356642269236, 11620141300950, 413852033403900, 14866630538956566, 581218522354753956, 25057946263703764470, 1130134186398074530860, 53379011394286882435446, 2696730444995933164241556

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

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

Mathematica

Table[(1/(n+1)!)* Sum[(3*k)!/k!*(n+k)!*StirlingS2[n, 3*k], {k, 0, Floor[n/3]}], {n, 0, 21}] (* Vincenzo Librandi, Feb 13 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-(exp(x)-1)^3))/x))
(Magma) [(1/Factorial(n+1)) * &+[Factorial(3*k) / Factorial(k) * Factorial(n+k)* StirlingSecond(n, 3*k): k in [0..Floor(n/3)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 13 2026

Crossrefs

Cf. A392765.

Sequence in context: A357087 A392850 A357025 * A357085 A224149 A055404

Adjacent sequences: A392766 A392767 A392768 * A392770 A392771 A392772

Keyword

nonn

Author

Seiichi Manyama, Jan 22 2026

Status

approved