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%I #14 Sep 21 2022 00:33:34
%S 1,2,24,968,128000,59973152,107564800000,774840978000000,
%T 22880602349081985024,2779532441951756299960832,
%U 1383517973370847653192530395136,2804745232514434754535719279455029248,23030042114303275003004737650852694413279232
%N a(n) = 2^n*(2^n + n)^(n-1).
%H G. C. Greubel, <a href="/A136524/b136524.txt">Table of n, a(n) for n = 0..57</a>
%F E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * W(2^n*x)^n * x^n/n!.
%F a(n) = coefficient of x^n/n! in W(x)^(2^n) where W(x) = LambertW(-x)/(-x).
%e E.g.f.: A(x) = 1 + 2*x + 24*x^2/2! + 968*x^3/3! + 128000*x^4/4! + ...
%e A(x) = 1 + 2*W(2x)*x + 2^4*W(4x)^2*x^2/2! + 2^9*W(8*x)^3*x^3/3! + ...
%e W(x) = LambertW(-x)/(-x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + … + (n+1)^(n-1)*x^n/n! + ...
%e This is a special application of the following identity.
%e Let F(x) be a formal power series in x such that F(0)=1, then
%e Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
%e Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b).
%e The e.g.f. of this sequence is derived as follows.
%e Let F(x) = W(x) = LambertW(-x)/(-x);
%e since log( W(x) ) = x*W(x) and
%e since W(x)^m = Sum_{n>=0} m*(m+n)^(n-1)*x^n/n! then
%e Sum_{n>=0} m^n * q^(n^2) * W(q^n*x)^(b+n) * x^n/ n! =
%e Sum_{n>=0} (m*q^n + b) * (m*q^n + b + n)^(n-1) * x^n.
%t Table[2^n*(2^n +n)^(n-1), {n,0,30}] (* _G. C. Greubel_, Sep 20 2022 *)
%o (PARI) {a(n)=local(W=sum(k=0,n,(k+1)^(k-1)*x^k/k!)); n!*polcoeff( (W+x*O(x^n))^(2^n), n)}
%o (Magma) [2^n*(2^n + n)^(n-1): n in [0..30]]; // _G. C. Greubel_, Sep 20 2022
%o (SageMath) [2^n*(2^n + n)^(n-1) for n in (0..30)] # _G. C. Greubel_, Sep 20 2022
%Y Cf. A136525.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Jan 03 2008
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