A097042 - OEIS

%I #24 Feb 21 2021 04:10:25

%S 1,2,0,4,2,4,4,8,8,10,12,16,20,24,28,36,42,48,60,72,84,100,116,136,

%T 160,186,216,252,292,336,388,448,512,588,672,768,878,1000,1136,1292,

%U 1464,1656,1876,2120,2388,2696,3032,3408,3832,4298,4816,5396,6036,6744,7532,8404

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

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

%C 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

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

%H Vaclav Kotesovec, <a href="/A097042/b097042.txt">Table of n, a(n) for n = 0..1000</a>

%H Min-Joo Jang, <a href="https://arxiv.org/abs/1703.01837">Asymptotic behavior of odd-even partitions</a>, arXiv:1703.01837v1 [math.NT], 2017.

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

%F 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

%t 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 *)

%Y Cf. A096661, A026832, A179049.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Sep 15 2004

%E Name corrected by _Peter Bala_, Feb 19 2021