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

0,2

Links

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

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

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 *)

Crossrefs

Column k=2 of A292622.

Sequence in context: A190660 A180158 A373014 * A173693 A058278 A097333

Adjacent sequences: A320686 A320687 A320688 * A320690 A320691 A320692

Keyword

nonn

Author

Alois P. Heinz, Oct 19 2018

Status

approved