A375394 - OEIS

%I #13 Aug 21 2024 11:22:10

%S 1,1,3,12,63,420,3345,31080,330225,3946320,52401195,765404640,

%T 12196214415,210533843520,3913845680745,77955813936000,

%U 1656235524168225,37387344753158400,893615568162592275,22545399132243187200,598744483093370188575,16696076277239091532800

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

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

%F a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / ((1 + LambertW(1)) * exp(n) * LambertW(1)^((n+1)/2)). - _Vaclav Kotesovec_, Aug 21 2024

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

%o (PARI) a(n) = n!*sum(k=0, n\2, ((n-2*k+1)/2)^k/k!);

%Y Cf. A072597, A375395.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 21 2024