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

1, 2, 2, 3, 5, 7, 10, 14, 19, 26, 35, 46, 61, 80, 103, 133, 171, 217, 275, 347, 435, 544, 677, 838, 1036, 1276, 1564, 1913, 2334, 2837, 3441, 4163, 5022, 6046, 7262, 8701, 10407, 12421, 14792, 17586, 20871, 24721, 29234, 34514, 40679, 47874, 56256, 66003

Offset: 0

Author

Alois P. Heinz, Oct 19 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Column k=2 of A292622.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      binomial(2, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);

Mathematica

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]];
a[n_] := b[n, n];
a /@ Range[0, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * sqrt(2) * n^(3/2)). - Vaclav Kotesovec, Oct 24 2018

G.f.: (1 + x)^2 * Product_{k>=2} 1 / (1 - x^k). - Ilya Gutkovskiy, Apr 24 2021