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%I #14 Oct 09 2019 01:23:11
%S 2,4,24,248,3600,67296,1538672,41593920,1297683360,45891815040,
%T 1814072216864,79263667304640,3793393788125760,197339219789611200,
%U 11087608251010390080,669127189486395204544,43167108189991530605184,2964541208087967215725440,215934869210274766223069440,16627513858173093851116296960,1349582577808759197056647917696,115158206188199564942934814336896,10305721256666828267464573643658240
%N G.f.: Sum_{n>=0} (1 + x)^(n*(n+1)/2) / 2^n.
%H Vaclav Kotesovec, <a href="/A265937/b265937.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: Sum_{n>=0} (1+x)^n/2^n * Product_{k=1..n} (2 - (1+x)^(2*k-1))/(2 - (1+x)^(2*k)) due to a q-series identity.
%F G.f.: 1/(1 - (1+x)/2 /(1 - (1+x)*((1+x)-1)/2 /(1 - (1+x)^3/2 /(1 - (1+x)^2*((1+x)^2-1)/2 /(1 - (1+x)^5/2 /(1 - (1+x)^3*((1+x)^3-1)/2 /(1 - (1+x)^7/2 /(1 - (1+x)^4*((1+x)^4-1)/2 /(1 - ...))))))))), a continued fraction due to a partial elliptic theta function identity.
%F a(n) = Sum_{k>=(sqrt(8*n+1)-1)/2} binomial(k*(k+1)/2,n) / 2^k.
%F a(n) = 2*A173219(n). - _Vaclav Kotesovec_, Oct 08 2019
%F a(n) ~ 2^(n+1) * n^n / (2^(log(2)/4) * log(2)^(2*n+1) * exp(n)). - _Vaclav Kotesovec_, Oct 08 2019
%e G.f.: A(x) = 2 + 4*x + 24*x^2 + 248*x^3 + 3600*x^4 + 67296*x^5 + 1538672*x^6 + 41593920*x^7 + 1297683360*x^8 + 45891815040*x^9 + 1814072216864*x^10 +...
%e where
%e A(x) = 1 + (1+x)/2 + (1+x)^3/2^2 + (1+x)^6/2^3 + (1+x)^10/2^4 + (1+x)^15/2^5 + (1+x)^21/2^6 + (1+x)^28/2^7 + (1+x)^36/2^8 +...+ (1+x)^(n*(n+1)/2)/2^n +...
%t Table[Sum[StirlingS1[n, j] * Sum[Binomial[j, s] * HurwitzLerchPhi[1/2, -j - s, 0], {s, 0, j}] / 2^j, {j, 0, n}] / n!, {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 08 2019 *)
%o (PARI) /* Informal listing of terms: */
%o {Vec( round( sum(n=0, 600, (1+x +O(x^31))^(n*(n+1)/2)/2^n * 1.) ) )}
%o {Vec( round( sum(n=0, 200, (1.+x)^n/2^n * prod(k=1, n, (2 - (1+x)^(2*k-1)) / (2 - (1+x)^(2*k)) +O(x^21) ) ) ) )}
%Y Cf. A265936.
%K nonn
%O 0,1
%A _Paul D. Hanna_, Dec 23 2015
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