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%I #7 Jul 20 2019 08:05:22
%S 1,3,6,12,22,38,64,104,164,254,386,576,848,1232,1768,2512,3534,4926,
%T 6812,9348,12736,17240,23192,31016,41256,54594,71890,94232,122976,
%U 159816,206872,266768,342756,438868,560064,712448,903526,1142478,1440528,1811384,2271720,2841800,3546224
%N Expansion of (1 + x) * Product_{k>=1} (1 + x^k)/(1 - x^k).
%F G.f.: (1 + x)/theta_4(x), where theta_4() is the Jacobi theta function.
%F a(n) = A015128(n) + A015128(n-1).
%F a(n) ~ exp(Pi*sqrt(n)) / (4*n) * (1 - (Pi/4 + 1/Pi)/sqrt(n)). - _Vaclav Kotesovec_, Jul 20 2019
%t nmax = 42; CoefficientList[Series[(1 + x) Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; Table[a[n] + a[n - 1], {n, 0, 42}]
%Y Cf. A001650, A015128, A052816, A084376, A207641, A211971, A277643.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jul 20 2019