A392887 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + log(1-x^2)^2/x) ).

1, 0, 0, 6, 0, 120, 2160, 4620, 322560, 4656960, 51408000, 2225805120, 36404121600, 935298524160, 34191689051520, 769760181590400, 29060360301772800, 1049180522277888000, 33363257180449996800, 1488084913957723361280, 58703745178055577600000

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 + log(1-(x*A(x))^2)^2/(x*A(x)).

a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} (2*(n-2*k))!/((n-2*k)! * (2*k+1)!) * |Stirling1(n-k,2*(n-2*k))|/(n-k)!.

Mathematica

Table[(n!)^2* Sum[(2*(n-2*k))!/((n-2*k)!*(2*k+1)!)*Abs[StirlingS1[n-k, 2*(n-2*k)]/(n-k)!], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vincenzo Librandi, Feb 04 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+log(1-x^2)^2/x))/x))
(Magma) [Factorial(n)^2 * &+[Factorial(2*(n - 2*k)) / (Factorial(n - 2*k) * Factorial(2*k + 1)) * Abs(StirlingFirst(n - k, 2*(n - 2*k)) / Factorial(n - k)): k in [0..Floor(n/2)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 04 2026

Crossrefs

Cf. A392852, A392888.

Sequence in context: A246137 A052679 A392891 * A392794 A392791 A266218

Adjacent sequences: A392884 A392885 A392886 * A392888 A392889 A392890

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved