A305105 G.f.: Sum_{k>=1} x^(2*k-1)/(1-x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

0, 1, 3, 8, 17, 34, 64, 114, 195, 325, 526, 832, 1292, 1970, 2958, 4384, 6413, 9276, 13283, 18836, 26478, 36924, 51096, 70210, 95844, 130019, 175350, 235192, 313802, 416618, 550540, 724250, 948719, 1237732, 1608508, 2082600, 2686857, 3454590, 4427144, 5655652

Offset: 0

Author

Vaclav Kotesovec, May 25 2018

Keywords

• nonn
• easy

Comments

Convolution of A066897 and A000009.

Convolution of A067588 and A000041.

Let A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 - x^k). Then A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) (mod 2). It follows from the comment in A001227 by Juri-Stepan Gerasimov, dated Jul 17 2016, that a(n) is odd iff n is a square or twice a square. - Peter Bala, Jan 10 2025

Links

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

Crossrefs

Cf. A001227, A015128, A066897, A067588, A305102, A305104.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^(2*k-1)/(1-x^(2*k-1)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (2*gamma + log(16*n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.