A302020 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^(2*k))/(1 - x^(2*k-1))).

1, 1, 2, 5, 12, 28, 66, 156, 367, 863, 2031, 4779, 11244, 26456, 62248, 146462, 344608, 810822, 1907769, 4488757, 10561519, 24850017, 58469179, 137571128, 323688747, 761601701, 1791959579, 4216270956, 9920391613, 23341519267, 54919860316, 129219997322, 304039515247, 715369360371

Offset

0,3

Links

Table of n, a(n) for n=0..33.

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Partition Function b_k

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(4*k))/(1 - x^k)).

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k))).

a(0) = 1; a(n) = Sum_{k=1..n} A001935(k-1)*a(n-k).

Mathematica

nmax = 33; CoefficientList[Series[1/(1 - x Product[(1 + x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 + (1 - x) QPochhammer[-1, x^2]/(2 QPochhammer[1/x, x^2])), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 - x EllipticTheta[2, 0, x]/(Sqrt[2] x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]])), {x, 0, nmax}], x]

Crossrefs

Antidiagonal sums of A296068.

Cf. A001935, A299106, A299108.

Sequence in context: A206721 A209173 A297496 * A239333 A255115 A166297

Adjacent sequences: A302017 A302018 A302019 * A302021 A302022 A302023

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 30 2018

Status

approved