A390217 Number of partitions of n with parts colored by {0, 1, 2, 3} such that the sum of colors is congruent to 0 (mod 4).

1, 1, 4, 10, 28, 63, 146, 310, 651, 1295, 2536, 4802, 8941, 16240, 29050, 50996, 88303, 150587, 253720, 422100, 694753, 1131190, 1824376, 2914730, 4617358, 7253925, 11309296, 17501344, 26897135, 41061345, 62291644, 93926230, 140814226, 209939415, 311348100

Offset

0,3

Comments

a(n) is the number of integer partitions of n colored by the coloring set {0,1,2,3} where the total color of a partition is taken to be the sum of the colors of each part congruent to 0 modulo 4.

Links

Sean A. Irvine, Table of n, a(n) for n = 0..56

Formula

Conjectured g.f.: (1/4) * (Product_{j>=1} 1/(1-x^j)^4 + Product_{j>=1} 1/(1-x^(2*j))^2 + 2*Product_{j>=1} 1/(1-x^(4*j))).

Conjectured g.f.: (1/4) * (G.f. of A023003(x) + G.f. of A000712(x^2) + 2*G.f. of A000041(x^4)).

Example

For n = 2 the 4 partitions of total color = 0 are (in the form (part, coloring)): (2, 0), (1, 0) + (1, 0), (1, 1) + (1, 3), (1, 2) + (1, 2).

Crossrefs

Cf. A000041, A109085, A281357, A390208, A023003, A000712, A389343, A387957.

Sequence in context: A032248 A092504 A340568 * A038579 A133726 A026534

Adjacent sequences: A390214 A390215 A390216 * A390218 A390219 A390220

Keyword

nonn

Author

Thomas Hutton, Oct 29 2025

Status

approved