A320689 - OEIS

%I #15 Apr 24 2021 09:55:19

%S 1,2,2,3,5,7,10,14,19,26,35,46,61,80,103,133,171,217,275,347,435,544,

%T 677,838,1036,1276,1564,1913,2334,2837,3441,4163,5022,6046,7262,8701,

%U 10407,12421,14792,17586,20871,24721,29234,34514,40679,47874,56256,66003

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

%H Alois P. Heinz, <a href="/A320689/b320689.txt">Table of n, a(n) for n = 0..10000</a>

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

%F G.f.: (1 + x)^2 * 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(2, 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[2, 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=2 of A292622.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 19 2018