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A320690 Number of partitions of n with up to three distinct kinds of 1.

%I A320690 #14 Apr 24 2021 09:55:26
%S A320690 1,3,4,5,8,12,17,24,33,45,61,81,107,141,183,236,304,388,492,622,782,
%T A320690 979,1221,1515,1874,2312,2840,3477,4247,5171,6278,7604,9185,11068,
%U A320690 13308,15963,19108,22828,27213,32378,38457,45592,53955,63748,75193,88553,104130
%N A320690 Number of partitions of n with up to three distinct kinds of 1.
%H A320690 Alois P. Heinz, <a href="/A320690/b320690.txt">Table of n, a(n) for n = 0..10000</a>
%F A320690 a(n) ~ Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2018
%F A320690 G.f.: (1 + x)^3 * Product_{k>=2} 1 / (1 - x^k). - _Ilya Gutkovskiy_, Apr 24 2021
%p A320690 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A320690       binomial(3, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))
%p A320690     end:
%p A320690 a:= n-> b(n$2):
%p A320690 seq(a(n), n=0..60);
%t A320690 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, Binomial[3, n], If[i > n, 0, b[n - i, i]] + b[n, i - 1]];
%t A320690 a[n_] := b[n, n];
%t A320690 a /@ Range[0, 60] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A320690 Column k=3 of A292622.
%K A320690 nonn
%O A320690 0,2
%A A320690 _Alois P. Heinz_, Oct 19 2018