A300415 Expansion of Product_{k>=2} (1 + x^k)/(1 - x^k).

1, 0, 2, 2, 4, 6, 10, 14, 22, 32, 46, 66, 94, 130, 182, 250, 340, 462, 622, 830, 1106, 1462, 1922, 2518, 3282, 4256, 5502, 7082, 9078, 11602, 14774, 18746, 23722, 29922, 37630, 47202, 59044, 73662, 91682, 113830, 140994, 174262, 214906, 264462, 324802, 398110, 487018, 594694

Offset: 0

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

• nonn

Comments

Convolution of the sequences A002865 and A025147.

Also number of overpartitions of n without a 1. - George Beck, Jan 25 2021

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A002865, A015128, A025147, A207641, A211971.

Programs

Maple

g:= (1-x)/((1+x)*JacobiTheta4(0,x)):
S:=series(g,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 05 2018

Mathematica

nmax = 47; CoefficientList[Series[Product[(1 + x^k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - x)/((1 + x) EllipticTheta[4, 0, x]), {x, 0, nmax}], x]

Formula

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

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

a(n) ~ Pi * exp(Pi*sqrt(n)) / (32*n^(3/2)). - Vaclav Kotesovec, Mar 05 2018