A285472 Expansion of Product_{k>=1} ((1 + x^(4*k)) / (1 - x^k)).

1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 106, 140, 192, 253, 342, 444, 588, 758, 992, 1267, 1634, 2072, 2650, 3334, 4218, 5276, 6627, 8234, 10262, 12682, 15708, 19308, 23764, 29070, 35597, 43340, 52792, 64008, 77622, 93724, 113160, 136124, 163712, 196225

Offset: 0

Author

Vaclav Kotesovec, Apr 19 2017

Keywords

• nonn

Comments

a(n) is the number of overpartitions wherein only parts that are a multiple of four may be overlined. - Alois P. Heinz, Feb 03 2025

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A006950, A015128, A279328, A266648, A285460.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 4)=0, 2, 1)*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..45);  # Alois P. Heinz, Feb 03 2025

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^(4*k))/(1-x^k)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ sqrt(3) * exp(sqrt(3*n)*Pi/2) / (16*n).