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

0,2

Links

Table of n, a(n) for n=0..49.

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.

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

Example

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].

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 *)

Crossrefs

Cf. A003105, A015128, A098151, A335755.

Sequence in context: A018550 A248475 A035952 * A105781 A035958 A035965

Adjacent sequences: A335751 A335752 A335753 * A335755 A335756 A335757

Keyword

nonn

Author

Jeremy Lovejoy, Jun 20 2020

Status

approved