A160976 Number of partitions of n where every part appears at least 6 times.

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 9, 6, 8, 9, 11, 9, 16, 12, 16, 16, 18, 18, 27, 21, 27, 28, 34, 31, 47, 39, 50, 50, 60, 57, 81, 72, 88, 89, 105, 101, 136, 124, 153, 151, 176, 171, 222, 205, 246, 252, 287, 281, 353, 334, 392, 401, 460, 453, 559, 534, 620, 636

Offset: 0

Author

R. H. Hardin, Jun 01 2009

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 1..1000 from R. H. Hardin)

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1), j=6..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..75);  # Alois P. Heinz, Feb 06 2024

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(6*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 28 2015 *)

Formula

a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-6*x)) dx = -1.055135119523138524396962100839537485211520908123400469186... . - Vaclav Kotesovec, Jan 05 2016

Extensions

a(0)=1 prepended by Seiichi Manyama, Feb 06 2024