A392794 - OEIS

%I #15 Feb 12 2026 17:33:43

%S 1,0,0,6,0,120,2880,2940,403200,8074080,47174400,3635866080,

%T 72648576000,1183289627520,71843629777920,1617274512453600,

%U 49788393887232000,2703881987142336000,74947823184858624000,3369543466000424993280,176517590187551348736000

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

%H Vincenzo Librandi, <a href="/A392794/b392794.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.

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

%t Table[(1/(n+1))* Sum[(2*(n-2*k))!/(n-2*k)!*(2*n-2*k)!*StirlingS2[n-k,2*(n-2*k)]/(n-k)!,{k,0,Floor[n/2]}],{n,0,21}] (* _Vincenzo Librandi_, Feb 12 2026 *)

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

%o (Magma) [(1/(n+1)) * &+[Factorial(2*(n-2*k))/Factorial(n-2*k) * Factorial(2*n-2*k)* StirlingSecond(n-k, 2*(n-2*k)) / Factorial(n-k): k in [0..Floor(n/2)]]: n in [0..25] ]; // _Vincenzo Librandi_, Feb 12 2026

%Y Cf. A392795, A392796.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jan 22 2026