A320690 - OEIS

%I #14 Apr 24 2021 09:55:26

%S 1,3,4,5,8,12,17,24,33,45,61,81,107,141,183,236,304,388,492,622,782,

%T 979,1221,1515,1874,2312,2840,3477,4247,5171,6278,7604,9185,11068,

%U 13308,15963,19108,22828,27213,32378,38457,45592,53955,63748,75193,88553,104130

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

%H Alois P. Heinz, <a href="/A320690/b320690.txt">Table of n, a(n) for n = 0..10000</a>

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

%F G.f.: (1 + x)^3 * 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(3, 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[3, 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, after _Alois P. Heinz_ *)

%Y Column k=3 of A292622.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 19 2018