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A265937
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G.f.: Sum_{n>=0} (1 + x)^(n*(n+1)/2) / 2^n.
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3
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2, 4, 24, 248, 3600, 67296, 1538672, 41593920, 1297683360, 45891815040, 1814072216864, 79263667304640, 3793393788125760, 197339219789611200, 11087608251010390080, 669127189486395204544, 43167108189991530605184, 2964541208087967215725440, 215934869210274766223069440, 16627513858173093851116296960, 1349582577808759197056647917696, 115158206188199564942934814336896, 10305721256666828267464573643658240
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OFFSET
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0,1
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LINKS
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FORMULA
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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.
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.
a(n) = Sum_{k>=(sqrt(8*n+1)-1)/2} binomial(k*(k+1)/2,n) / 2^k.
a(n) ~ 2^(n+1) * n^n / (2^(log(2)/4) * log(2)^(2*n+1) * exp(n)). - Vaclav Kotesovec, Oct 08 2019
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EXAMPLE
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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 +...
where
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 +...
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MATHEMATICA
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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 *)
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PROG
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(PARI) /* Informal listing of terms: */
{Vec( round( sum(n=0, 600, (1+x +O(x^31))^(n*(n+1)/2)/2^n * 1.) ) )}
{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) ) ) ) )}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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