A320690 Number of partitions of n with up to three distinct kinds of 1.

1, 3, 4, 5, 8, 12, 17, 24, 33, 45, 61, 81, 107, 141, 183, 236, 304, 388, 492, 622, 782, 979, 1221, 1515, 1874, 2312, 2840, 3477, 4247, 5171, 6278, 7604, 9185, 11068, 13308, 15963, 19108, 22828, 27213, 32378, 38457, 45592, 53955, 63748, 75193, 88553, 104130

Offset

0,2

Links

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

Formula

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

G.f.: (1 + x)^3 * 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(3, 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[3, 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=3 of A292622.

Sequence in context: A355158 A034403 A215082 * A179070 A358356 A368800

Adjacent sequences: A320687 A320688 A320689 * A320691 A320692 A320693

Keyword

nonn

Author

Alois P. Heinz, Oct 19 2018

Status

approved