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A265936 G.f.: Sum_{n>=0} (1 + x)^(n^2) / 2^n.

%I A265936 #37 Oct 08 2019 09:11:24
%S A265936 2,6,72,1488,43212,1615824,73897824,3995603040,249332628600,
%T A265936 17635891224600,1394325697514112,121850733102557184,
%U A265936 11663364820483368384,1213527023075625127296,136368036713802512640384,16459661773011642351224832,2123742016843422531580031760,291703805646180152870305600416,42495460119330209128505618419584,6544578588779477399509681497008256,1062399800520315889891506552001161024,181308080907736435566683700136306288320
%N A265936 G.f.: Sum_{n>=0} (1 + x)^(n^2) / 2^n.
%H A265936 Vaclav Kotesovec, <a href="/A265936/b265936.txt">Table of n, a(n) for n = 0..330</a>
%F A265936 G.f.: Sum_{n>=0} (1+x)^n/2^n * Product_{k=1..n} (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)) due to a q-series identity.
%F A265936 G.f.: 1/(1 - (1+x)/2 /(1 - (1+x)*((1+x)^2-1)/2 /(1 - (1+x)^5/2 /(1 - (1+x)^3*((1+x)^4-1)/2 /(1 - (1+x)^9/2 /(1 - (1+x)^5*((1+x)^6-1)/2 /(1 - (1+x)^13/2 /(1 - (1+x)^7*((1+x)^8-1)/2 /(1 - ...))))))))), a continued fraction due to a partial elliptic theta function identity.
%F A265936 a(n) = Sum_{k>=sqrt(n)} binomial(k^2,n) / 2^k.
%F A265936 a(n) = Sum_{k=0..2*n} A303920(n,k) * 2^k, for n>0.
%F A265936 a(n) = 2 * A173217(n) for n>=0.
%F A265936 a(n) ~ 2^(2*n + 1/2 - log(2)/8) * n^n / (exp(n) * log(2)^(2*n + 1)). - _Vaclav Kotesovec_, Oct 08 2019
%e A265936 G.f.: A(x) = 2 + 6*x + 72*x^2 + 1488*x^3 + 43212*x^4 + 1615824*x^5 + 73897824*x^6 + 3995603040*x^7 + 249332628600*x^8 + 17635891224600*x^9 +...
%e A265936 where
%e A265936 A(x) = 1 + (1+x)/2 + (1+x)^4/2^2 + (1+x)^9/2^3 + (1+x)^16/2^4 + (1+x)^25/2^5 + (1+x)^36/2^6 + (1+x)^49/2^7 + (1+x)^64/2^8 +...+ (1+x)^(n^2)/2^n +...
%t A265936 Table[Round[Sum[Binomial[k^2, n]/2^k, {k, Sqrt[n], Infinity}]] , {n, 0, 20}] (* _G. C. Greubel_, May 23 2017 *)
%t A265936 Table[2*Sum[StirlingS1[n, j] * HurwitzLerchPhi[1/2, -2*j, 0]/2, {j, 0, n}] / n!, {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 08 2019 *)
%o A265936 (PARI) /* Informal listing of terms: */
%o A265936 {Vec( round( sum(n=0,600,(1+x +O(x^31))^(n^2)/2^n * 1.) ) )}
%o A265936 {Vec( round( sum(n=0,200, (1.+x)^n/2^n * prod(k=1,n, (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)) +O(x^21) ) ) ) )}
%Y A265936 Cf. A265937, A303920, A173217.
%K A265936 nonn
%O A265936 0,1
%A A265936 _Paul D. Hanna_, Dec 23 2015