A320691 Number of partitions of n with up to four distinct kinds of 1.

1, 4, 7, 9, 13, 20, 29, 41, 57, 78, 106, 142, 188, 248, 324, 419, 540, 692, 880, 1114, 1404, 1761, 2200, 2736, 3389, 4186, 5152, 6317, 7724, 9418, 11449, 13882, 16789, 20253, 24376, 29271, 35071, 41936, 50041, 59591, 70835, 84049, 99547, 117703, 138941, 163746

Offset

0,2

Links

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

Formula

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

G.f.: (1 + x)^4 * 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(4, 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[4, 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=4 of A292622.

Sequence in context: A310962 A310963 A082869 * A139444 A353020 A166569

Adjacent sequences: A320688 A320689 A320690 * A320692 A320693 A320694

Keyword

nonn

Author

Alois P. Heinz, Oct 19 2018

Status

approved