A173219
G.f.: A(x) = Sum_{n>=0} (1 + x)^(n(n+1)/2) / 2^(n+1).
Original entry on oeis.org
1, 2, 12, 124, 1800, 33648, 769336, 20796960, 648841680, 22945907520, 907036108432, 39631833652320, 1896696894062880, 98669609894805600, 5543804125505195040, 334563594743197602272, 21583554094995765302592
Offset: 0
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Table[Sum[StirlingS1[n, j] * Sum[Binomial[j, s]*HurwitzLerchPhi[1/2, -j - s, 0], {s, 0, j}] / 2^(j+1), {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2019 *)
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{a(n)=local(A=sum(m=0,n^2+100,(1+x +O(x^(n+2)))^(m*(m+1)/2)/2^(m+1)));round(polcoeff(A,n))}
A265936
G.f.: Sum_{n>=0} (1 + x)^(n^2) / 2^n.
Original entry on oeis.org
2, 6, 72, 1488, 43212, 1615824, 73897824, 3995603040, 249332628600, 17635891224600, 1394325697514112, 121850733102557184, 11663364820483368384, 1213527023075625127296, 136368036713802512640384, 16459661773011642351224832, 2123742016843422531580031760, 291703805646180152870305600416, 42495460119330209128505618419584, 6544578588779477399509681497008256, 1062399800520315889891506552001161024, 181308080907736435566683700136306288320
Offset: 0
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 +...
where
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 +...
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Table[Round[Sum[Binomial[k^2, n]/2^k, {k, Sqrt[n], Infinity}]] , {n, 0, 20}] (* G. C. Greubel, May 23 2017 *)
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 *)
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/* Informal listing of terms: */
{Vec( round( sum(n=0,600,(1+x +O(x^31))^(n^2)/2^n * 1.) ) )}
{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) ) ) ) )}
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