A392991 - OEIS

%I #16 Feb 02 2026 10:16:29

%S 1,0,0,0,0,120,0,5040,0,211680,18144000,9979200,5748019200,536215680,

%T 1322204083200,45801294739200,278970531840000,48375730910707200,

%U 58341630393446400,32935581898547328000,705879485427916800000,18775935637141716633600,1733749973244641544192000

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

%H Vincenzo Librandi, <a href="/A392991/b392991.txt">Table of n, a(n) for n = 0..300</a>

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

%F 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!.

%t 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 *)

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

%o (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

%Y Cf. A392891, A392957.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Jan 29 2026