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

0,3

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.

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

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 *)

Crossrefs

Cf. A056970.

Sequence in context: A307704 A139716 A168232 * A216960 A285348 A361008

Adjacent sequences: A253133 A253134 A253135 * A253137 A253138 A253139

Keyword

nonn

Author

Jeremy Lovejoy, Mar 23 2015

Status

approved