A320692 Number of partitions of n with up to five distinct kinds of 1.

1, 5, 11, 16, 22, 33, 49, 70, 98, 135, 184, 248, 330, 436, 572, 743, 959, 1232, 1572, 1994, 2518, 3165, 3961, 4936, 6125, 7575, 9338, 11469, 14041, 17142, 20867, 25331, 30671, 37042, 44629, 53647, 64342, 77007, 91977, 109632, 130426, 154884, 183596, 217250

Offset

0,2

Links

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

Formula

a(n) ~ Pi * 2^(5/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - Vaclav Kotesovec, Oct 24 2018

G.f.: (1 + x)^5 * 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(5, 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[5, 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 *)

Crossrefs

Column k=5 of A292622.

Sequence in context: A314169 A299911 A314170 * A314171 A314172 A314173

Adjacent sequences: A320689 A320690 A320691 * A320693 A320694 A320695

Keyword

nonn

Author

Alois P. Heinz, Oct 19 2018

Status

approved