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.

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, 90, 101, 116, 129, 146, 162, 182, 203, 228, 254, 284, 314, 351, 388, 433, 478, 531, 584, 646, 711, 785, 863, 952, 1044, 1149, 1258, 1384, 1513, 1660, 1812, 1983, 2163

Offset: 0

Author

Jeremy Lovejoy, Jun 21 2020

Keywords

• nonn

Examples

			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].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• 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. A253144.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, i-1), j=0..min(n/i, [0, n, 1, 0, 1, 0][irem(i, 6)+1]))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 21 2020

Mathematica

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]]]}]]];
a[n_] := b[n, n];
a /@ Range[0, 60] (* Jean-François Alcover, Dec 01 2020, after Alois P. Heinz *)
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 *)

Formula

G.f.: Product_{k>=1} (1+q^(6*k-2))*(1+q^(6*k-4))/(1-q^(6*k-5)).

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