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%I #9 Oct 10 2024 06:53:40
%S 1,0,2,2,2,2,6,6,10,10,14,14,26,26,38,46,58,66,86,94,130,146,182,214,
%T 274,306,382,438,530,602,750,838,1018,1162,1390,1598,1898,2154,2550,
%U 2910,3402,3858,4550,5134,5970,6786,7846,8902,10306,11618,13390,15142,17346,19562,22398
%N G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j).
%C In general, if d >= 1, b > 0 and g.f. = Sum_{k>=0} d^k * x^(b*k^2 + c*k) / Product_{j=1..k} (1 - x^j), then a(n) ~ r^c * (b*log(r)^2 + polylog(2, 1-r))^(1/4) * exp(2*sqrt((b*log(r)^2 + polylog(2, 1-r))*n)) / (2*sqrt((2*b*(1-r) + r)*Pi) * n^(3/4)), where r is the smallest positive real root of the equation d*r^(2*b) + r = 1.
%H Vaclav Kotesovec, <a href="/A376947/b376947.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) ~ (Pi^2/6 + log(2)^2)^(1/4) * exp(sqrt((Pi^2/3 + 2*log(2)^2)*n)) / (2^(7/4) * sqrt(3*Pi) * n^(3/4)).
%t nmax = 80; CoefficientList[Series[Sum[2^k*x^(k*(k+1)) / Product[1-x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%Y Cf. A003106, A032302, A376943, A376948.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Oct 10 2024