A253136 The number of overpartitions of n into parts congruent to 2, 4, or 5 modulo 6.

1, 0, 2, 0, 4, 2, 6, 4, 10, 8, 18, 14, 28, 24, 44, 42, 68, 66, 102, 104, 154, 160, 226, 238, 330, 354, 476, 516, 676, 742, 958, 1056, 1342, 1486, 1862, 2076, 2568, 2872, 3516, 3940, 4782, 5370, 6464, 7268, 8686, 9774, 11606, 13070, 15428, 17380, 20408, 22986

Offset: 0

Author

Jeremy Lovejoy, Mar 23 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• J. Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory 1 (2005), 215-224.

Crossrefs

Cf. A056970.

Programs

Maple

series(mul((1+x^(6*k+2))*(1+x^(6*k+4))*(1+x^(6*k+5))/((1-x^(6*k+2))*(1-x^(6*k+4))*(1-x^(6*k+5))), k=0..100), x=0, 100);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
     `if`(irem(i, 6) in [2, 4, 5], add(2*b(n-i*j, i-1), j=1..n/i), 0)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 04 2019

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[ MemberQ[{2, 4, 5}, Mod[i, 6]], Sum[2b[n - i j, i-1], {j, 1, n/i}], 0]]];
a[n_] := b[n, n];
a /@ Range[0, 60] (* Jean-François Alcover, Nov 23 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)) / ((1 - x^(6*k+2)) * (1 - x^(6*k+4)) * (1 - x^(6*k+5))), {k, 0, nmax/6}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

Formula

a(n) ~ Pi^(5/6) * exp(Pi*sqrt(n/2)) / (2^(7/4) * 3^(1/6) * Gamma(1/6) * n^(11/12)). - Vaclav Kotesovec, Jan 14 2021