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a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^n.
6

%I #11 May 07 2018 09:31:21

%S 1,1,-1,-5,-1,31,65,-90,-641,-644,3329,11386,-1471,-87021,-164634,

%T 317935,1881471,1418719,-11370760,-33937951,17468929,294971868,

%U 468897758,-1304743033,-6275603903,-2804572819,42665919997,109181454826,-106020803386,-1063546684834,-1362993953395

%N a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^n.

%H G. C. Greubel, <a href="/A296043/b296043.txt">Table of n, a(n) for n = 0..500</a>

%t Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 + x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]

%t Table[SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^n, {x, 0, n}], {n, 0, 30}]

%Y Main diagonal of A296067.

%Y Cf. A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845, A296044, A296045.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Dec 03 2017