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A320689 Number of partitions of n with up to two distinct kinds of 1.

%I A320689 #15 Apr 24 2021 09:55:19
%S A320689 1,2,2,3,5,7,10,14,19,26,35,46,61,80,103,133,171,217,275,347,435,544,
%T A320689 677,838,1036,1276,1564,1913,2334,2837,3441,4163,5022,6046,7262,8701,
%U A320689 10407,12421,14792,17586,20871,24721,29234,34514,40679,47874,56256,66003
%N A320689 Number of partitions of n with up to two distinct kinds of 1.
%H A320689 Alois P. Heinz, <a href="/A320689/b320689.txt">Table of n, a(n) for n = 0..10000</a>
%F A320689 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * sqrt(2) * n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2018
%F A320689 G.f.: (1 + x)^2 * Product_{k>=2} 1 / (1 - x^k). - _Ilya Gutkovskiy_, Apr 24 2021
%p A320689 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A320689       binomial(2, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))
%p A320689     end:
%p A320689 a:= n-> b(n$2):
%p A320689 seq(a(n), n=0..60);
%t A320689 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, Binomial[2, n], If[i > n, 0, b[n - i, i]] + b[n, i - 1]];
%t A320689 a[n_] := b[n, n];
%t A320689 a /@ Range[0, 60] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A320689 Column k=2 of A292622.
%K A320689 nonn
%O A320689 0,2
%A A320689 _Alois P. Heinz_, Oct 19 2018