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

Author

Jeremy Lovejoy, Jun 20 2020

Keywords

• nonn

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.

Crossrefs

Cf. A003105, A015128, A098151, A335755.

Programs

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