A392828 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^2*(exp(x) - 1)^3) ).

1, 0, 0, 0, 0, 120, 1080, 6300, 30240, 130032, 18665640, 660623700, 13658934960, 212523761736, 2764009560312, 77520900775500, 4099129923470400, 172565711699473632, 5540231695274913096, 145530177163066645380, 3989207890319572270800, 168566608848038865057720

Offset

0,6

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)^3.

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/5)} (3*k)! * binomial(n+1,k) * Stirling2(n-2*k,3*k)/(n-2*k)!.

Mathematica

Table[(n!/(n+1))* Sum[(3*k)!*Binomial[n+1, k]*StirlingS2[n-2*k, 3*k]/(n-2*k)!, {k, 0, Floor[n/5]}], {n, 0, 21}] (* Vincenzo Librandi, Feb 06 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\5, (3*k)!*binomial(n+1, k)*stirling(n-2*k, 3*k, 2)/(n-2*k)!)/(n+1);
(Magma) [(Factorial(n)/(n+1)) * &+[Factorial(3*k)* Binomial(n+1, k) * StirlingSecond(n - 2*k, 3*k) / Factorial(n - 2*k): k in [0..Floor(n/5)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 06 2026

Crossrefs

Cf. A371139, A392827.

Cf. A392832.

Sequence in context: A262387 A133119 A052777 * A052765 A392832 A265091

Adjacent sequences: A392825 A392826 A392827 * A392829 A392830 A392831

Keyword

nonn

Author

Seiichi Manyama, Jan 24 2026

Status

approved