A335730 - OEIS

%I #12 Jun 20 2020 09:38:06

%S 2,4,12,20,40,72,124,200,330,520,804,1224,1832,2704,3960,5704,8144,

%T 11532,16164,22480,31056,42568,57972,78480,105610,141336,188208,

%U 249352,328824,431760,564468,734992,953424,1232144,1586760,2036580,2605352,3322584,4224624,5355920

%N a(n) is the number of smallest parts in the overpartitions of n having odd smallest part.

%H S. Ahlgren, K. Bringmann, and J. Lovejoy, <a href="https://doi.org/10.1016/j.aim.2011.05.024">l-adic properties of smallest parts functions</a>, Advances in Mathematics, 228 (2011), 629-645.

%H K. Bringmann, J. Lovejoy, and R. Osburn, <a href="https://doi.org/10.1016/j.jnt.2008.10.017">Rank and crank moments for overpartitions</a>, Journal of Number Theory, 129 (2009), 1758-1772.

%H K. Bringmann, J. Lovejoy, and R. Osburn, <a href="https://doi.org/10.1093/imrn/rnp131">Automorphic properties of generating functions for generalized rank moments and Durfee symbols</a>, International Mathematics Research Notices, (2010), 238-260.

%F a(n) = A335724(n) - A335728(n).

%F G.f.: (Product_{k>=1} (1+q^k)/(1-q^k))*(Sum_{n>=1} 2*n*q^n/(1-q^(2*n)) + Sum_{n=-oo..oo, n<>0} 4*(-1)^n*q^(n^2+n)*(1+q^(2*n)+q^(3*n))/((1-q^(2*n))*(1-q^(4*n)))).

%e There are 14 overpartitions of 4: [4], [4'], [3,1], [3,1'], [3',1], [3',1'], [2,2], [2',2], [2,1,1], [2,1',1], [2',1,1], [2',1',1], [1,1,1,1], [1',1,1,1], and so a(4) = 20.

%Y Cf. A015128, A092269, A235792, A335724, A335728.

%K nonn

%O 1,1

%A _Jeremy Lovejoy_, Jun 19 2020