A207944 Expansion of Product_{i>=1} (1 + x^(2*i + 1))/(1 - x^(2*i + 1)).

1, 0, 0, 2, 0, 2, 2, 2, 4, 4, 6, 6, 10, 10, 14, 18, 20, 26, 32, 38, 46, 58, 66, 82, 98, 116, 138, 166, 194, 230, 274, 318, 376, 442, 514, 602, 704, 814, 950, 1102, 1274, 1474, 1706, 1962, 2262, 2606, 2986, 3430, 3934, 4496, 5144, 5878, 6698, 7638, 8698, 9886

Offset: 0

Author

Roger L. Bagula, Feb 21 2012

Keywords

• nonn

References

• George E. Andrews, Number Theory, Dover Publications, N.Y., 1971, 164-165.
• Samuel I. Goldberg, Curvature and Homology, Dover, New York, 1998, 144.

Links

• Shi-Chao Chen, On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32--37. MR3181230. The g.f. occurs on page 32, by accident (the product there should really start at n=0, not 1, in order to give A080054!). - _N. J. A. Sloane_, Apr 24 2014

Crossrefs

Cf. A015128, A080054, A142724.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k + 1))/(1 - x^(2*k + 1)), {k, 1, nmax}], {x, 0, nmax}], x] (* fixed by Vaclav Kotesovec, Apr 13 2017 *)

Formula

a(n) ~ exp(sqrt(n/2)*Pi) * Pi / (2^(19/4) * n^(5/4)). - Vaclav Kotesovec, Apr 13 2017