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A320692 Number of partitions of n with up to five distinct kinds of 1.

%I A320692 #13 Apr 24 2021 20:44:13
%S A320692 1,5,11,16,22,33,49,70,98,135,184,248,330,436,572,743,959,1232,1572,
%T A320692 1994,2518,3165,3961,4936,6125,7575,9338,11469,14041,17142,20867,
%U A320692 25331,30671,37042,44629,53647,64342,77007,91977,109632,130426,154884,183596,217250
%N A320692 Number of partitions of n with up to five distinct kinds of 1.
%H A320692 Alois P. Heinz, <a href="/A320692/b320692.txt">Table of n, a(n) for n = 0..10000</a>
%F A320692 a(n) ~ Pi * 2^(5/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2018
%F A320692 G.f.: (1 + x)^5 * Product_{k>=2} 1 / (1 - x^k). - _Ilya Gutkovskiy_, Apr 24 2021
%p A320692 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A320692       binomial(5, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))
%p A320692     end:
%p A320692 a:= n-> b(n$2):
%p A320692 seq(a(n), n=0..60);
%t A320692 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, Binomial[5, n], If[i > n, 0, b[n - i, i]] + b[n, i - 1]];
%t A320692 a[n_] := b[n, n];
%t A320692 a /@ Range[0, 60] (* _Jean-François Alcover_, Dec 14 2020 *)
%Y A320692 Column k=5 of A292622.
%K A320692 nonn
%O A320692 0,2
%A A320692 _Alois P. Heinz_, Oct 19 2018