A392890 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + (exp(x^3) - 1)/x^2) ).

1, 1, 2, 6, 36, 420, 6120, 94080, 1545600, 28667520, 620676000, 15462770400, 427269427200, 12769595879040, 409975771645440, 14163639025536000, 526670239685990400, 20988647304174950400, 890661296247564441600, 40028284097655549696000, 1898798845826870507520000

Offset

0,3

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))^3) - 1)/(x*A(x))^2.

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

Mathematica

Table[(n!)^2*Sum[1/(3*k+1)!*Abs[StirlingS2[n-2*k, n-3*k]/(n-2*k)!], {k, 0, Floor[n/3]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 26 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+(exp(x^3)-1)/x^2))/x))
(Magma) [ Factorial(n)^2 * &+[1 / Factorial(3*k + 1) * Abs(StirlingSecond(n - 2*k, n - 3*k) / Factorial(n - 2*k)): k in [0..Floor(n/3)]] : n in [0..25] ]; // Vincenzo Librandi, Jan 26 2026

Crossrefs

Cf. A000272, A392889.

Sequence in context: A262234 A371043 A055512 * A392886 A078973 A208650

Adjacent sequences: A392887 A392888 A392889 * A392891 A392892 A392893

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved