A392831 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^2*log(1-x)^2) ).

1, 0, 0, 0, 24, 120, 660, 4200, 191968, 3519936, 53760240, 792506880, 22388914944, 669140334720, 18170020552320, 462130871529600, 14328176733941760, 520857435665710080, 19497450954605783040, 712110759540855091200, 27605997143525828044800, 1182427775256949482885120

Offset

0,5

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

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/4)} (2*k)! * binomial(n+1,k) * |Stirling1(n-2*k,2*k)|/(n-2*k)!.

Mathematica

Table[(n!/(n+1))* Sum[(2*k)!*Binomial[n+1, k]*Abs[StirlingS1[n-2*k, 2*k]]/(n-2*k)!, {k, 0, Floor[n/3]}], {n, 0, 21}] (* Vincenzo Librandi, Feb 07 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\4, (2*k)!*binomial(n+1, k)*abs(stirling(n-2*k, 2*k, 1))/(n-2*k)!)/(n+1);
(Magma) [(Factorial (n)/(n+1)) * &+[Factorial(2*k)* Binomial(n+1, k)* Abs(StirlingFirst(n - 2*k, 2*k)) / Factorial(n -2*k): k in [0..Floor(n/3)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 07 2026

Crossrefs

Cf. A371138, A392832.

Sequence in context: A179720 A235702 A052754 * A392761 A050213 A124657

Adjacent sequences: A392828 A392829 A392830 * A392832 A392833 A392834

Keyword

nonn

Author

Seiichi Manyama, Jan 24 2026

Status

approved