A097042 G.f. = (1 + 4 * g.f. for A096661)/(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)).

1, 2, 0, 4, 2, 4, 4, 8, 8, 10, 12, 16, 20, 24, 28, 36, 42, 48, 60, 72, 84, 100, 116, 136, 160, 186, 216, 252, 292, 336, 388, 448, 512, 588, 672, 768, 878, 1000, 1136, 1292, 1464, 1656, 1876, 2120, 2388, 2696, 3032, 3408, 3832, 4298, 4816, 5396, 6036, 6744, 7532, 8404

Offset: 0

Author

N. J. A. Sloane, Sep 15 2004

Keywords

• nonn

Comments

a(0) = 1; for n>0, a(n) = 2*A026832(n) (i.e., essentially Fine's numbers L(n) multiplied by 2).

The number of odd-even overpartitions of n: an odd-even overpartition of n is an overpartition of n with the smallest part odd and such that the difference between successive parts is odd if the smaller part is nonoverlined and even otherwise - see Yang 2017. - Peter Bala, Mar 29 2017

References

• N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Min-Joo Jang, Asymptotic behavior of odd-even partitions, arXiv:1703.01837v1 [math.NT], 2017.

Crossrefs

Cf. A096661, A026832, A179049.

Programs

Mathematica

nmax = 60; Flatten[{1, Rest[CoefficientList[Series[2*Sum[x^(2*k - 1) QPochhammer[-x^(2*k), x], {k, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Mar 28 2017 *)

Formula

a(n) ~ 1/(3^(5/4)*n^(3/4))*exp(Pi*sqrt(n/3)) [Jang 2017]. - Peter Bala, Mar 29 2017

Conjectural g.f.: 1 + 2*Sum_{n >= 1} q^(n*(n+1)/2)/( (1 + q^n) * Product_{k = 1..n} 1 - q^k ). - Peter Bala, Feb 19 2021

Extensions

Name corrected by Peter Bala, Feb 19 2021