A392002 - OEIS

%I #11 Jan 01 2026 17:33:09

%S 0,1,2,5,9,18,31,55,90,150,237,376,577,885,1325,1978,2900,4235,6100,

%T 8745,12400,17501,24477,34075,47079,64756,88493,120420,162940,219595,

%U 294476,393407,523237,693465,915384,1204329,1578702,2063035,2686950,3489365,4517456,5832448,7508754,9641915

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

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

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

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

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

%e 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).

%Y Cf. A390208, A387957, A000712, A000041.

%K nonn

%O 0,3

%A _Thomas Hutton_, Dec 26 2025