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A320691 Number of partitions of n with up to four distinct kinds of 1.

%I A320691 #14 Apr 24 2021 20:44:07
%S A320691 1,4,7,9,13,20,29,41,57,78,106,142,188,248,324,419,540,692,880,1114,
%T A320691 1404,1761,2200,2736,3389,4186,5152,6317,7724,9418,11449,13882,16789,
%U A320691 20253,24376,29271,35071,41936,50041,59591,70835,84049,99547,117703,138941,163746
%N A320691 Number of partitions of n with up to four distinct kinds of 1.
%H A320691 Alois P. Heinz, <a href="/A320691/b320691.txt">Table of n, a(n) for n = 0..10000</a>
%F A320691 a(n) ~ Pi * 2^(3/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2018
%F A320691 G.f.: (1 + x)^4 * Product_{k>=2} 1 / (1 - x^k). - _Ilya Gutkovskiy_, Apr 24 2021
%p A320691 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A320691       binomial(4, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))
%p A320691     end:
%p A320691 a:= n-> b(n$2):
%p A320691 seq(a(n), n=0..60);
%t A320691 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, Binomial[4, n], If[i > n, 0, b[n - i, i]] + b[n, i - 1]];
%t A320691 a[n_] := b[n, n];
%t A320691 a /@ Range[0, 60] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A320691 Column k=4 of A292622.
%K A320691 nonn
%O A320691 0,2
%A A320691 _Alois P. Heinz_, Oct 19 2018