A309266 Expansion of (1 + x) * Product_{k>=1} (1 + x^k)/(1 - x^k).

1, 3, 6, 12, 22, 38, 64, 104, 164, 254, 386, 576, 848, 1232, 1768, 2512, 3534, 4926, 6812, 9348, 12736, 17240, 23192, 31016, 41256, 54594, 71890, 94232, 122976, 159816, 206872, 266768, 342756, 438868, 560064, 712448, 903526, 1142478, 1440528, 1811384, 2271720, 2841800, 3546224

Offset: 0

Author

Ilya Gutkovskiy, Jul 20 2019

Keywords

• nonn

Crossrefs

Cf. A001650, A015128, A052816, A084376, A207641, A211971, A277643.

Programs

Mathematica

nmax = 42; CoefficientList[Series[(1 + x) Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; Table[a[n] + a[n - 1], {n, 0, 42}]

Formula

G.f.: (1 + x)/theta_4(x), where theta_4() is the Jacobi theta function.

a(n) = A015128(n) + A015128(n-1).

a(n) ~ exp(Pi*sqrt(n)) / (4*n) * (1 - (Pi/4 + 1/Pi)/sqrt(n)). - Vaclav Kotesovec, Jul 20 2019