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A335766 a(n) is the number of partitions of n into parts congruent to 1, 2, or 4 modulo 6 where only parts congruent to 1 modulo 6 may be repeated.

%I A335766 #20 Jan 14 2021 11:54:06
%S A335766 1,1,2,2,3,3,4,5,6,7,9,10,12,14,18,20,24,27,31,35,41,47,54,61,70,78,
%T A335766 90,101,116,129,146,162,182,203,228,254,284,314,351,388,433,478,531,
%U A335766 584,646,711,785,863,952,1044,1149,1258,1384,1513,1660,1812,1983,2163
%N A335766 a(n) is the number of partitions of n into parts congruent to 1, 2, or 4 modulo 6 where only parts congruent to 1 modulo 6 may be repeated.
%H A335766 Alois P. Heinz, <a href="/A335766/b335766.txt">Table of n, a(n) for n = 0..10000</a>
%H A335766 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 A335766 G.f.: Product_{k>=1} (1+q^(6*k-2))*(1+q^(6*k-4))/(1-q^(6*k-5)).
%F A335766 a(n) ~ Gamma(1/6) * exp(sqrt(2*n)*Pi/3) / (2^(23/12) * sqrt(3) * Pi^(5/6) * n^(7/12)). - _Vaclav Kotesovec_, Jan 14 2021
%e A335766 a(8) = 6, the relevant partitions being [8], [7,1], [4,2,1,1], [4,1,1,1,1],[2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
%p A335766 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A335766       b(n-i*j, i-1), j=0..min(n/i, [0, n, 1, 0, 1, 0][irem(i, 6)+1]))))
%p A335766     end:
%p A335766 a:= n-> b(n$2):
%p A335766 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jun 21 2020
%t A335766 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1], {j, 0, Min[n/i, {0, n, 1, 0, 1, 0}[[Mod[i, 6] + 1]]]}]]];
%t A335766 a[n_] := b[n, n];
%t A335766 a /@ Range[0, 60] (* _Jean-François Alcover_, Dec 01 2020, after _Alois P. Heinz_ *)
%t A335766 nmax = 60; CoefficientList[Series[Product[(1 + x^(6*k-2)) * (1 + x^(6*k-4)) / (1 - x^(6*k-5)), {k, 1, nmax/6}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 14 2021 *)
%Y A335766 Cf. A253144.
%K A335766 nonn
%O A335766 0,3
%A A335766 _Jeremy Lovejoy_, Jun 21 2020