A335730 -id:A335730 - cp's OEIS frontend

A335724 a(n) is the number of smallest parts in the overpartitions of n.

2, 6, 12, 26, 44, 84, 136, 230, 366, 580, 884, 1356, 2012, 2968, 4320, 6226, 8856, 12522, 17508, 24324, 33528, 45892, 62392, 84372, 113374, 151548, 201552, 266752, 351380, 460920, 601992, 783158, 1014984, 1310600, 1686408, 2162814, 2764748, 3523324, 4476720, 5671748

Offset: 1

Jeremy Lovejoy, Jun 19 2020

nonn

			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) = 26.

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.

Cf. A015128, A092269, A235792, A335728, A335730.

G.f.: 2*(Product_{k>=1} (1+q^k)/(1-q^k))*Sum_{n>=1} (q^n*Product_{j=1..n}(1-q^j))/((1-q^n)^2*Product_{j=1..n}(1+q^j)).

a(n) = A335728(n) + A335730(n).

A335728 a(n) is the number of smallest parts in the overpartitions of n having even smallest part.

Original entry on oeis.org

0, 2, 0, 6, 4, 12, 12, 30, 36, 60, 80, 132, 180, 264, 360, 522, 712, 990, 1344, 1844, 2472, 3324, 4420, 5892, 7764, 10212, 13344, 17400, 22556, 29160, 37524, 48166, 61560, 78456, 99648, 126234, 159396, 200740, 252096, 315828

Offset: 1

Jeremy Lovejoy, Jun 19 2020

nonn

			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) = 6.

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.

Cf. A015128, A092269, A235792, A335724, A335730.

G.f.: 2*(Product_{k>=1} (1+q^k)/(1-q^k))*Sum_{n>=1} (q^(2*n)*Product_{j=1..2*n}(1-q^j))/((1-q^(2*n))^2*Product_{j=1..2*n}(1+q^j)).

a(n) = A335724(n) - A335730(n).

cp's OEIS Frontend

A335724 a(n) is the number of smallest parts in the overpartitions of n.

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