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%I #14 May 22 2023 16:41:40
%S 1,1,0,-8,-38,106,3676,24508,-296036,-9149156,-56500064,2211573376,
%T 64958496472,184823374360,-35372361487280,-971135892546224,
%U 4364710018963216,1034808592156017424,25290798052846014208,-474242641154857953152,-49625273567646267051104
%N E.g.f. satisfies A(x) = exp(x - x^2/2 * A(x)^2).
%H Robert Israel, <a href="/A362492/b362492.txt">Table of n, a(n) for n = 0..403</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f.: exp(x - LambertW(x^2 * exp(2*x))/2) = sqrt( LambertW(x^2 * exp(2*x))/x^2 ).
%F a(n) = n! * Sum_{k=0..floor(n/2)} (-1/2)^k * (2*k+1)^(n-k-1) / (k! * (n-2*k)!).
%p N:= 50: # for a(0)..a(N)
%p egf:= exp(x - LambertW(x^2 * exp(2*x))/2):
%p S:=series(egf,x,N+1):
%p [seq](coeff(S,x,i)*i!,i=0..N); # _Robert Israel_, May 22 2023
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^2*exp(2*x))/2)))
%Y Cf. A362493, A362494.
%Y Cf. A362480.
%K sign
%O 0,4
%A _Seiichi Manyama_, Apr 22 2023
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