A327717 Expansion of Product_{k>=1} (1 + x^k/(1 + x^(2*k))).

1, 1, 1, 1, 2, 3, 2, 3, 5, 6, 6, 7, 10, 12, 12, 15, 20, 23, 24, 28, 36, 42, 44, 51, 64, 73, 78, 89, 108, 123, 132, 150, 179, 202, 218, 246, 288, 324, 350, 393, 456, 509, 552, 616, 706, 786, 852, 948, 1078, 1195, 1297, 1436, 1620, 1791, 1942, 2145, 2406, 2650, 2874, 3163, 3528

Offset: 0

Author

Seiichi Manyama, Sep 23 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Convolution inverse of A307757.

Cf. A000726, A327716, A327718, A327719, A327720.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[1 + x^k/(1 + x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k + x^(2*k)) * (1 - x^(4*k - 2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

PARI

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1+x^(2*k))))

Formula

a(n) ~ 5^(1/4) * exp(sqrt(5*n/2)*Pi/3) / (2^(5/4)*3*n^(3/4)). - Vaclav Kotesovec, Sep 23 2019