A392850 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + (exp(x) - 1)^3) ).

1, 0, 0, 6, 36, 150, 2700, 54726, 771876, 12945750, 327177180, 8394923526, 209893127796, 6106875462870, 203608601845740, 7013022801220806, 253941767834096196, 10074938765228850390, 427741076842456307580, 18976040774569773250566, 888872049703600193539476

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A392828, A392851.

Sequence in context: A353774 A357010 A357087 * A357025 A392769 A357085

Adjacent sequences: A392847 A392848 A392849 * A392851 A392852 A392853

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved