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A371038 E.g.f. satisfies A(x) = exp(x^2*A(x)) / (1-x).

%I A371038 #20 Mar 09 2025 12:40:37
%S A371038 1,1,4,18,132,1140,12720,164640,2514960,43500240,850076640,
%T A371038 18418609440,439831909440,11457415569600,323707663319040,
%U A371038 9854548934630400,321709145793235200,11209975693710393600,415330670608805952000,16303720885477254028800
%N A371038 E.g.f. satisfies A(x) = exp(x^2*A(x)) / (1-x).
%H A371038 Robert Israel, <a href="/A371038/b371038.txt">Table of n, a(n) for n = 0..399</a>
%H A371038 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A371038 E.g.f.: LambertW( -x^2/(1-x) ) / (-x^2).
%F A371038 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * binomial(n-k,n-2*k)/k!.
%F A371038 a(n) ~ exp(2) * sqrt(1 + 4*exp(1) - sqrt(1 + 4*exp(1))) * 2^(n + 3/2) * n^(n-1) / ((1 + 2*exp(1) - sqrt(1 + 4*exp(1)))*(-1 + sqrt(1 + 4*exp(1)))^(n+1)). - _Vaclav Kotesovec_, Mar 12 2024
%p A371038 E:=  LambertW( -x^2/(1-x) ) / (-x^2):
%p A371038 S:= series(E,x,43):
%p A371038 seq(coeff(S,x,i)*i!,i=0..40); # _Robert Israel_, Mar 09 2025
%o A371038 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(-x^2/(1-x))/(-x^2)))
%o A371038 (PARI) a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*binomial(n-k, n-2*k)/k!);
%Y A371038 Cf. A352410, A371039.
%K A371038 nonn
%O A371038 0,3
%A A371038 _Seiichi Manyama_, Mar 09 2024