A168481 - OEIS

%I #5 Nov 05 2014 13:53:53

%S 1,4,56,2432,377600,222691328,513752956928,4690384533848064,

%T 170085542794237050880,24520078828632712041988096,

%U 14055876186039467842015007342592

%N G.f.: Sum_{n>=0} (n+1)*2^(n^2)*(1 + 2^n*x)^n*x^n.

%C This sequence illustrates the identity:

%C Sum_{n>=0} (n+1)*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))^2.

%F a(n) = [x^n] 1/(1 - 2^n*x*(1+x))^2.

%F a(n) = Sum_{k=0..[n/2]} (n-k+1)*C(n-k,k)*2^(n(n-k)).

%F a(n) ~ n * 2^(n^2). - _Vaclav Kotesovec_, Nov 05 2014

%e G.f.: A(x) = 1 + 4*x + 56*x^2 + 2432*x^3 + 377600*x^4 +...

%t Table[Sum[(n-k+1)*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,(m+1)*(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))^2,n)}

%o (PARI) {a(n)=sum(k=0,n\2,(n-k+1)*binomial(n-k,k)*2^(n*(n-k)))}

%Y Cf. A168480, A168482.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 26 2009