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%I #5 Oct 07 2020 08:33:07
%S 1,6,108,5888,1107456,768540672,2038400811008,21012301788217344,
%T 848380447466572480512,134688875052459668084359168,
%U 84279244669071810947388769566720
%N G.f.: Sum_{n>=0} [(n+1)(n+2)/2]*2^(n^2)*(1 + 2^n*x)^n*x^n.
%C This sequence illustrates the identity:
%C Sum_{n>=0} [(n+1)(n+2)/2]*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))^3.
%F a(n) = [x^n] 1/(1 - 2^n*x*(1+x))^3.
%F a(n) = Sum_{k=0..[n/2]} [(n-k+1)(n-k+1)/2]*C(n-k,k)*2^(n(n-k)).
%F a(n) ~ n^2 * 2^(n^2 - 1). - _Vaclav Kotesovec_, Oct 07 2020
%e G.f.: A(x) = 1 + 6*x + 108*x^2 + 5888*x^3 + 1107456*x^4 +...
%t Table[Sum[(n - k + 1)*(n - k + 2)/2 * Binomial[n - k, k]*2^(n*(n - k)), {k, 0, n/2}], {n, 0, 15}] (* _Vaclav Kotesovec_, Oct 07 2020 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,(m+1)*(m+2)/2*(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))^3,n)}
%o (PARI) {a(n)=sum(k=0,n\2,(n-k+1)*(n-k+2)/2*binomial(n-k,k)*2^(n*(n-k)))}
%Y Cf. A168480, A168481.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 26 2009
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