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A335730
a(n) is the number of smallest parts in the overpartitions of n having odd smallest part.
2
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
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
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Jun 19 2020
STATUS
approved