A392992 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*(exp(x^3) - 1)^2) ).

1, 0, 0, 0, 0, 0, 0, 5040, 0, 0, 3628800, 0, 0, 3632428800, 610248038400, 0, 5230697472000, 6046686277632000, 0, 10474994757427200, 52712876843827200000, 3576365952019660800000, 28100018194440192000, 495496987495295385600000, 171243758878374085263360000

Offset

0,8

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))^3) - 1)^2.

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

Mathematica

Table[(n!)^2* Sum[(2*(n-3*k))!/((n-3*k)!*(3*k+1)!) *Abs[StirlingS2[k, 2*(n-3*k)]]/k!, {k, 0, Floor[n/3]}], {n, 0, 24}] (* Vincenzo Librandi, Feb 01 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x*(exp(x^3)-1)^2))/x))
(Magma) [Factorial(n)^2* &+[Factorial(2*(n-3*k))/(Factorial(n-3*k) * Factorial(3*k+1)) * StirlingSecond(k, 2*(n-3*k))/Factorial(k): k in [0..Floor(n/3)] ] : n in [0..24] ]; // Vincenzo Librandi, Feb 01 2026

Crossrefs

Cf. A392892, A392996.

Sequence in context: A309034 A237690 A269125 * A392994 A111030 A068378

Adjacent sequences: A392989 A392990 A392991 * A392993 A392994 A392995

Keyword

nonn

Author

Seiichi Manyama, Jan 29 2026

Status

approved