A335754 a(n) is the number of overpartitions of n where overlined parts are not divisible by 3 and non-overlined parts are congruent to 1 modulo 3.
1, 2, 3, 4, 6, 9, 12, 17, 23, 30, 39, 51, 66, 84, 107, 135, 168, 209, 259, 319, 391, 478, 581, 703, 849, 1022, 1226, 1466, 1748, 2078, 2465, 2917, 3443, 4055, 4765, 5588, 6540, 7640, 8908, 10368, 12047, 13973, 16182, 18712, 21604, 24906, 28673, 32964, 37846, 43397
Offset: 0
Keywords
Examples
The 9 overpartitions counted by a(5) are: [5'], [4,1], [4,1'], [4',1], [4',1'], [2',1,1,1], [2',1',1,1], [1,1,1,1,1], [1',1,1,1,1].
Links
- J. Lovejoy, Asymmetric generalizations of Schur's theorem, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.
Programs
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Mathematica
nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k-1)) * (1 + x^(3*k-2)) / (1 - x^(3*k-2)), {k, 1, nmax/3}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)
Formula
G.f.: Product_{n>=1} (1+q^(3*n-1))*(1+q^(3*n-2))/(1-q^(3*n-2)).
a(n) ~ Gamma(1/3) * exp(2*Pi*sqrt(n)/3) / (2^(3/2) * sqrt(3) * Pi^(2/3) * n^(2/3)). - Vaclav Kotesovec, Jan 14 2021