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

0,3

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.

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

Example

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].

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 *)

Crossrefs

Cf. A253144.

Sequence in context: A003114 A185227 A217569 * A026823 A025148 A096749

Adjacent sequences: A335763 A335764 A335765 * A335767 A335768 A335769

Keyword

nonn

Author

Jeremy Lovejoy, Jun 21 2020

Status

approved