A363045 Number of partitions of n whose greatest part is a multiple of 3.

1, 0, 0, 1, 1, 2, 4, 5, 7, 11, 14, 19, 27, 34, 45, 60, 77, 99, 130, 163, 208, 265, 333, 417, 526, 651, 810, 1004, 1237, 1519, 1869, 2278, 2780, 3382, 4101, 4958, 5995, 7210, 8669, 10398, 12444, 14859, 17730, 21086, 25057, 29718, 35186, 41584, 49100, 57842, 68075, 79991

Offset

0,6

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Formula

G.f.: Sum_{k>=0} x^(3*k)/Product_{j=1..3*k} (1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n) * (1 - (1/Pi + Pi/72)*sqrt(3/(2*n))). - Vaclav Kotesovec, May 20 2023

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-3*i, min(n-3*i, 3*i)), i=0..n/3):
seq(a(n), n=0..60);  # Alois P. Heinz, May 14 2023

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := Sum[b[n - 3*i, Min[n - 3*i, 3*i]], {i, 0, n/3}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 23 2023, after Alois P. Heinz *)

Prog

(PARI) my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k)/prod(j=1, 3*k, 1-x^j)))

Crossrefs

Column 3 of A363048.

Cf. A038499, A072233.

Sequence in context: A373030 A097697 A035620 * A022439 A050134 A010065

Adjacent sequences: A363042 A363043 A363044 * A363046 A363047 A363048

Keyword

nonn

Author

Seiichi Manyama, May 14 2023

Status

approved