A390208 Number of integer partitions of n with parts colored by {0, 1, 2} such that the sum of colors is congruent to 1 (mod 3).

0, 1, 3, 7, 17, 36, 73, 143, 270, 492, 880, 1533, 2621, 4403, 7281, 11858, 19074, 30294, 47586, 73970, 113883, 173727, 262820, 394407, 587480, 868868, 1276479, 1863360, 2703782, 3900666, 5596698, 7988198, 11344797, 16034671, 22559703, 31600179, 44076648, 61228998, 84723444, 116790329, 160408600

Offset

0,3

Comments

Equivalently, the number of such partitions congruent to 2 (mod 3).

The sequence is the 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 n modulo 3, n coprime to 3.

Links

Sean A. Irvine, Table of n, a(n) for n = 0..72

Formula

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

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

Example

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

Crossrefs

Cf. A000716, A000041, A389343.

Sequence in context: A260677 A014395 A326520 * A295146 A014314 A221792

Adjacent sequences: A390205 A390206 A390207 * A390209 A390210 A390211

Keyword

nonn

Author

Thomas Hutton, Oct 29 2025

Status

approved