A363046 Number of partitions of n whose greatest part is a multiple of 4.

1, 0, 0, 0, 1, 1, 2, 3, 6, 7, 11, 14, 21, 26, 36, 45, 62, 76, 100, 124, 162, 199, 255, 314, 399, 488, 612, 748, 932, 1134, 1400, 1699, 2086, 2520, 3072, 3700, 4488, 5384, 6494, 7766, 9326, 11112, 13283, 15778, 18788, 22245, 26386, 31150, 36825, 43345, 51070, 59953

Offset: 0

Author

Seiichi Manyama, May 14 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column 4 of A363048.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
a:= n-> add(b(n-4*i, min(n-4*i, 4*i)), i=0..n/4):
seq(a(n), n=0..60);  # Alois P. Heinz, May 14 2023

PARI

my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(4*k)/prod(j=1, 4*k, 1-x^j)))

Formula

G.f.: Sum_{k>=0} x^(4*k)/Product_{j=1..4*k} (1-x^j).

a(n) ~ A000041(n)/4. - Vaclav Kotesovec, May 21 2023