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%I #15 May 03 2015 07:16:44
%S 1,1,3,16,118,1116,12869,175096,2745726,48756438,967026762,
%T 21188546616,508286084222,13249410224210,372908807794347,
%U 11270832179901016,364083312029454453,12518063823862065816,456432182550333723335,17591590487681007523476
%N G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n^2).
%H Vaclav Kotesovec, <a href="/A220352/b220352.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: Sum_{n>=1} ((1+x)^n - 1)^(n-1) / (1+x)^(n^2).
%F a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 1.44832302735058524286860126583754380692... . - _Vaclav Kotesovec_, Nov 08 2014
%F In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1+log(2))/2)). - _Vaclav Kotesovec_, May 03 2015
%e G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1116*x^5 + 12869*x^6 +...
%e where the g.f. satisfies the identities:
%e (1) A(x) = 1 + x/(1+x) + (2*x + x^2)^2/(1+x)^4 + (3*x + 3*x^2 + x^3)^3/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^4/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^5/(1+x)^25 +...
%e (2) A(x) = 1/(1+x) + (2*x + x^2)/(1+x)^4 + (3*x + 3*x^2 + x^3)^2/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^3/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^4/(1+x)^25 +...
%o (PARI) {a(n)=local(q=1+x+x*O(x^n),A=1);A=sum(k=0,n,q^(-k^2)*(q^k-1)^k);polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n)=local(q=1+x+x*O(x^n),A=1);A=sum(k=1,n+1,q^(-k^2)*(q^k-1)^(k-1));polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A220353, A121886, A187826.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 11 2012
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