A392002 Number of partitions of n with parts colored by {0, 1} such that the sum of colors is congruent to 1 (mod 2).

0, 1, 2, 5, 9, 18, 31, 55, 90, 150, 237, 376, 577, 885, 1325, 1978, 2900, 4235, 6100, 8745, 12400, 17501, 24477, 34075, 47079, 64756, 88493, 120420, 162940, 219595, 294476, 393407, 523237, 693465, 915384, 1204329, 1578702, 2063035, 2686950, 3489365, 4517456, 5832448, 7508754, 9641915

Offset

0,3

Comments

a(n) is the number of integer partitions of n colored by the coloring set {0, 1} where the sum of colors are odd.

Links

Table of n, a(n) for n=0..43.

Formula

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

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

Example

For n = 2 the 2 partitions of total color = 1 are (in the form (part, coloring)): (2, 1), (1, 0) + (1, 1).
For n = 3 the 5 partitions of total color = 1 are (in the form (part, coloring)): (3, 1), (2, 1) + (1, 0), (2, 0) + (1, 1), (1, 1) + (1, 1) + (1, 1), (1, 0) + (1, 0) + (1, 1).

Crossrefs

Cf. A390208, A387957, A000712, A000041.

Sequence in context: A278690 A335838 A335837 * A288578 A002883 A295724

Adjacent sequences: A391999 A392000 A392001 * A392003 A392004 A392005

Keyword

nonn

Author

Thomas Hutton, Dec 26 2025

Status

approved