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a(n) is the number of overpartitions of n where overlined parts are not divisible by 3 and non-overlined parts are congruent to 2 modulo 3.
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%I #13 Jan 14 2021 11:50:18

%S 1,1,2,2,3,5,6,9,11,14,19,24,31,39,48,61,75,93,114,139,169,205,248,

%T 298,358,428,510,607,719,851,1005,1182,1389,1628,1904,2225,2592,3015,

%U 3501,4058,4698,5429,6264,7216,8302,9538,10944,12541,14351,16403

%N a(n) is the number of overpartitions of n where overlined parts are not divisible by 3 and non-overlined parts are congruent to 2 modulo 3.

%H J. Lovejoy, <a href="https://doi.org/10.1007/978-3-319-68376-8_26">Asymmetric generalizations of Schur's theorem</a>, 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.

%F G.f.: Product_{n>=1} (1+q^(3*n-1))*(1+q^(3*n-2))/(1-q^(3*n-1)).

%F a(n) ~ Pi^(2/3) * exp(2*Pi*sqrt(n)/3) / (3*sqrt(2)*Gamma(1/3)*n^(5/6)). - _Vaclav Kotesovec_, Jan 14 2021

%e The 6 overpartitions counted by a(6) are: [5,1'], [5',1'], [4',2], [4',2'], [2,2,2], [2',2,2].

%t nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k-1)) * (1 + x^(3*k-2)) / (1 - x^(3*k-1)), {k, 1, nmax/3}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 14 2021 *)

%Y Cf. A003105, A015128, A098151, A335754.

%K nonn

%O 0,3

%A _Jeremy Lovejoy_, Jun 20 2020