A392991 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*(exp(x^2) - 1)^2) ).

1, 0, 0, 0, 0, 120, 0, 5040, 0, 211680, 18144000, 9979200, 5748019200, 536215680, 1322204083200, 45801294739200, 278970531840000, 48375730910707200, 58341630393446400, 32935581898547328000, 705879485427916800000, 18775935637141716633600, 1733749973244641544192000

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)*(exp((x*A(x))^2) - 1)^2.

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

Mathematica

Table[(n!)^2*Sum[(2*(n-2*k))!/((n-2*k)!*(2*k+1)!)*StirlingS2[k, 2*(n-2*k)]/k!, {k, 0, Floor[n/2]}], {n, 0, 18}] (* Vincenzo Librandi, Feb 01 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x*(exp(x^2)-1)^2))/x))
(Magma) [Factorial(n)^2* &+[Factorial(2*(n-2*k))/(Factorial(n-2*k) * Factorial(2*k+1)) * StirlingSecond(k, 2*(n-2*k))/Factorial(k): k in [0..Floor(n/2)] ] : n in [0..22] ]; // Vincenzo Librandi, Feb 01 2026

Crossrefs

Cf. A392891, A392957.

Sequence in context: A242836 A229031 A392935 * A392993 A221406 A267428

Adjacent sequences: A392988 A392989 A392990 * A392992 A392993 A392994

Keyword

nonn

Author

Seiichi Manyama, Jan 29 2026

Status

approved