A375540 - OEIS

%I #8 Sep 02 2024 01:48:31

%S 1,2,12,126,1880,36250,856212,23928758,772172592,28253043378,

%T 1155731972780,52265163565582,2589097062756360,139428505876012106,

%U 8110011431007355716,506710228437429986790,33844577422630735656032,2406541293179536265812834,181497377154154817667851100

%N a(n) = 2^n * n! * [x^n] (1/2 - exp(-x))^n.

%F a(n) ~ n^n / (sqrt(1+LambertW(-exp(-1)/2)) * exp(n) * (-LambertW(-exp(-1)/2))^n). - _Vaclav Kotesovec_, Sep 01 2024

%p gf := n -> (1/2 - exp(-x))^n:

%p ser := n -> series(gf(n), x, 20):

%p a := n -> expand(2^n*n!*coeff(ser(n), x, n)):

%p seq(a(n), n = 0..18);

%t Table[2^n * n! * SeriesCoefficient[(1/2 - E^(-x))^n,{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 01 2024 *)

%K nonn

%O 0,2

%A _Peter Luschny_, Sep 01 2024