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A375540 a(n) = 2^n * n! * [x^n] (1/2 - exp(-x))^n.

%I A375540 #25 Aug 04 2025 09:27:44
%S A375540 1,2,12,126,1880,36250,856212,23928758,772172592,28253043378,
%T A375540 1155731972780,52265163565582,2589097062756360,139428505876012106,
%U A375540 8110011431007355716,506710228437429986790,33844577422630735656032,2406541293179536265812834,181497377154154817667851100
%N A375540 a(n) = 2^n * n! * [x^n] (1/2 - exp(-x))^n.
%H A375540 Alois P. Heinz, <a href="/A375540/b375540.txt">Table of n, a(n) for n = 0..362</a>
%H A375540 Sangchul Lee, <a href="https://math.stackexchange.com/a/5086916/929945">Fast and simple recursive algorithm for A375540</a>, answer to question on Mathematics Stack Exchange, Jul 31 2025.
%F A375540 a(n) ~ n^n / (sqrt(1+LambertW(-exp(-1)/2)) * exp(n) * (-LambertW(-exp(-1)/2))^n). - _Vaclav Kotesovec_, Sep 01 2024
%p A375540 gf := n -> (1/2 - exp(-x))^n:
%p A375540 ser := n -> series(gf(n), x, 20):
%p A375540 a := n -> expand(2^n*n!*coeff(ser(n), x, n)):
%p A375540 seq(a(n), n = 0..18);
%p A375540 # second Maple program:
%p A375540 b:= proc(n, k) option remember;
%p A375540      `if`(n=0, 1, k*(b(n-1, k)+b(n-1, k-1)))
%p A375540     end:
%p A375540 a:= n-> b(n$2):
%p A375540 seq(a(n), n=0..18);  # _Alois P. Heinz_, Aug 03 2025
%t A375540 Table[2^n * n! * SeriesCoefficient[(1/2 - E^(-x))^n,{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 01 2024 *)
%Y A375540 Cf. A002720, A195979.
%K A375540 nonn
%O A375540 0,2
%A A375540 _Peter Luschny_, Sep 01 2024