A304627 - OEIS

%I #13 Feb 16 2025 08:33:54

%S 1,0,2,6,12,22,38,62,98,152,230,342,502,726,1038,1470,2060,2862,3946,

%T 5398,7334,9902,13286,17726,23526,31064,40822,53406,69566,90246,

%U 116622,150142,192610,246254,313806,398638,504884,637590,802934,1008446,1263270,1578526,1967694,2447062

%N a(n) = [x^n] Product_{k>=1} (1 + x^k)*(1 - x^(n*k))/((1 - x^k)*(1 + x^(n*k))).

%H Vaclav Kotesovec, <a href="/A304627/b304627.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: 1/theta_4(x) - 2*x/(1 - x), where theta_4() is the Jacobi theta function.

%F a(n) ~ exp(Pi*sqrt(n)) / (8*n). - _Vaclav Kotesovec_, May 19 2018

%t Table[SeriesCoefficient[Product[(1 + x^k) (1 - x^(n k))/((1 - x^k) (1 + x^(n k))) , {k, 1, n}], {x, 0, n}], {n, 0, 43}]

%t Table[SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n - 1}], {x, 0, n}], {n, 0, 43}]

%t Join[{1}, Table[SeriesCoefficient[EllipticTheta[4, 0, x^n]/EllipticTheta[4, 0, x], {x, 0, n}], {n, 43}]]

%t nmax = 43; CoefficientList[Series[1/EllipticTheta[4, 0, x] - 2 x/(1 - x), {x, 0, nmax}], x]

%Y Cf. A000065, A015128, A052839, A080054, A098151, A103258, A111133, A138526, A262968.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 15 2018