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

%I A160990 #14 Feb 06 2024 18:02:10
%S A160990 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,
%T A160990 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,
%U A160990 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
%N A160990 Number of partitions of n where every part appears at least 20 times.
%H A160990 Alois P. Heinz, <a href="/A160990/b160990.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..1000 from R. H. Hardin)
%F A160990 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
%p A160990 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A160990       b(n, i-1)+add(b(n-i*j, i-1), j=20..n/i)))
%p A160990     end:
%p A160990 a:= n-> b(n$2):
%p A160990 seq(a(n), n=0..108);  # _Alois P. Heinz_, Feb 06 2024
%t A160990 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 *)
%K A160990 nonn
%O A160990 0,41
%A A160990 _R. H. Hardin_, Jun 01 2009
%E A160990 a(0)=1 prepended by _Alois P. Heinz_, Feb 06 2024