A335730 a(n) is the number of smallest parts in the overpartitions of n having odd smallest part.

2, 4, 12, 20, 40, 72, 124, 200, 330, 520, 804, 1224, 1832, 2704, 3960, 5704, 8144, 11532, 16164, 22480, 31056, 42568, 57972, 78480, 105610, 141336, 188208, 249352, 328824, 431760, 564468, 734992, 953424, 1232144, 1586760, 2036580, 2605352, 3322584, 4224624, 5355920

Offset

1,1

Links

Table of n, a(n) for n=1..40.

S. Ahlgren, K. Bringmann, and J. Lovejoy, l-adic properties of smallest parts functions, Advances in Mathematics, 228 (2011), 629-645.

K. Bringmann, J. Lovejoy, and R. Osburn, Rank and crank moments for overpartitions, Journal of Number Theory, 129 (2009), 1758-1772.

K. Bringmann, J. Lovejoy, and R. Osburn, Automorphic properties of generating functions for generalized rank moments and Durfee symbols, International Mathematics Research Notices, (2010), 238-260.

Formula

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

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)))).

Example

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.

Crossrefs

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

Sequence in context: A056228 A166869 A168380 * A308286 A297184 A218871

Adjacent sequences: A335727 A335728 A335729 * A335731 A335732 A335733

Keyword

nonn

Author

Jeremy Lovejoy, Jun 19 2020

Status

approved