A320692 - OEIS

%I #13 Apr 24 2021 20:44:13

%S 1,5,11,16,22,33,49,70,98,135,184,248,330,436,572,743,959,1232,1572,

%T 1994,2518,3165,3961,4936,6125,7575,9338,11469,14041,17142,20867,

%U 25331,30671,37042,44629,53647,64342,77007,91977,109632,130426,154884,183596,217250

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

%H Alois P. Heinz, <a href="/A320692/b320692.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ Pi * 2^(5/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2018

%F G.f.: (1 + x)^5 * Product_{k>=2} 1 / (1 - x^k). - _Ilya Gutkovskiy_, Apr 24 2021

%p b:= proc(n, i) option remember; `if`(n=0 or i=1,

%p       binomial(5, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..60);

%t 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 a[n_] := b[n, n];

%t a /@ Range[0, 60] (* _Jean-François Alcover_, Dec 14 2020 *)

%Y Column k=5 of A292622.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 19 2018