A389343 Number of integer partitions of n with parts colored by {0, 1, 2} such that the sum of colors is congruent to 0 (mod 3).

1, 1, 3, 8, 17, 36, 75, 143, 270, 495, 880, 1533, 2626, 4403, 7281, 11865, 19074, 30294, 47597, 73970, 113883, 173742, 262820, 394407, 587502, 868868, 1276479, 1863390, 2703782, 3900666, 5596740, 7988198, 11344797, 16034727, 22559703, 31600179, 44076725, 61228998, 84723444, 116790430, 160408600

Offset

0,3

Comments

a(n) is the number of integer partitions of n colored by the coloring set {0,1,2} where the total color of a partition is taken to be the sum of the colors of each part congruent to 0 modulo 3.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Formula

Conjectured g.f.: (1/3) * (Product_{j>=1} 1/(1-x^j)^3 + 2*Product_{j>=1} 1/(1-x^(3*j))).

Conjectured g.f.: (1/3) * (G.f. of A000716(x) + 2*G.f. of A000041(x^3)).

Example

For n = 2 the 3 partitions of total color = 0 are (in the form (part, coloring)): (2, 0), (1, 0) + (1, 0), (1, 1) + (1, 2).

Crossrefs

Cf. A000716, A000041, A109085, A281357, A390208.

Sequence in context: A106691 A140176 A238496 * A097391 A202554 A034481

Adjacent sequences: A389340 A389341 A389342 * A389344 A389345 A389346

Keyword

nonn

Author

Thomas Hutton, Oct 29 2025

Status

approved