A160990 Number of partitions of n where every part appears at least 20 times.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 7, 7, 8, 7, 10, 8, 10, 10, 11, 10, 13, 11, 15, 14, 15, 15, 19, 16, 19, 19, 21, 20, 23, 21, 25, 24, 25, 25, 29, 26, 29, 29, 34, 31, 35, 33, 38, 38, 39, 38, 44

Offset: 0

Author

R. H. Hardin, Jun 01 2009

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 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=20..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..108);  # Alois P. Heinz, Feb 06 2024

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(20*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(-20*x)) dx = -1.354168532835449099374593344112387373408094711414623392193... . - Vaclav Kotesovec, Jan 05 2016

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 06 2024