A392891 - OEIS

%I #18 Jan 30 2026 09:53:33

%S 1,0,0,6,0,120,2160,2940,322560,4445280,39312000,2198861280,

%T 31135104000,769192744320,32350774135680,597288505413600,

%U 24495937449984000,919567111913049600,24828597683118028800,1240480507369945121280,47003467274263019520000,1804660990139653802956800

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

%H Vincenzo Librandi, <a href="/A392891/b392891.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) satisfies A(x) = 1 + (exp((x*A(x))^2) - 1)^2/(x*A(x)).

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

%t Table[(n!)^2*Sum[(2*(n-2*k))!/((n-2*k)!*(2*k+1)!)*Abs[StirlingS2[n-k,2*(n-2*k)]/(n-k)!],{k,0,Floor[n/2]}],{n,0,23}] (* _Vincenzo Librandi_, Jan 26 2026 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+(exp(x^2)-1)^2/x))/x))

%o (Magma) [ Factorial(n)^2 * &+[Factorial(2*(n - 2*k)) / (Factorial(n - 2*k) * Factorial(2*k + 1)) * Abs(StirlingSecond(n - k, 2*(n - 2*k)) / Factorial(n - k)): k in [0..Floor(n/2)]]: n in [0..25] ]; // _Vincenzo Librandi_, Jan 26 2026

%Y Cf. A392849, A392892.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jan 25 2026