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%I #5 Nov 05 2014 13:52:53
%S 1,2,20,640,78080,37847040,74189111296,589682903613440,
%T 18955380356036952064,2455824622368881511497728,
%U 1278825951842748707166092263424,2671459568763422966186162922297753600
%N G.f.: Sum_{n>=0} 2^(n^2)*(1 + 2^n*x)^n*x^n.
%C This sequence illustrates the identity:
%C Sum_{n>=0} q^(n^2)*G(q^n*x)^n*x^n = Sum_{n>=0} c(n)*x^n
%C where c(n) = [x^n] 1/(1 - q^n*x*G(x)).
%F a(n) = [x^n] 1/(1 - 2^n*x*(1+x)).
%F a(n) = Sum_{k=0..[n/2]} C(n-k,k)*2^(n(n-k)).
%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Nov 05 2014
%e G.f.: A(x) = 1 + 2*x + 20*x^2 + 640*x^3 + 78080*x^4 +...
%t Table[Sum[Binomial[n-k,k]*2^(n*(n-k)),{k,0,Floor[n/2]}],{n,0,15}] (* _Vaclav Kotesovec_, Nov 05 2014 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,(1+2^m*x)^m*2^(m^2)*x^m)+x*O(x^n),n)}
%o (PARI) {a(n)=polcoeff(1/(1-2^n*x*(1+x)+x*O(x^n)),n)}
%o (PARI) {a(n)=sum(k=0,n\2,binomial(n-k,k)*2^(n*(n-k)))}
%Y Cf. A168481, A168482.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 26 2009
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