The sequence enumerating order-ideals in P_{n+1,n+2} that alternate colors, \

    with coloring parameter c=0, up to the , 23, -th term  is



[2, 4, 7, 17, 30, 80, 143, 404, 728, 2140, 3876, 11729, 21318, 65952, 120175,

    378321, 690690, 2205168, 4032015, 13023324, 23841480, 77761008, 142498692]



The sequence enumerating order-ideals in P_{n+1,n+2} that alternate colors, \

    with coloring parameter c=1, up to the , 23, -th term  is



[2, 3, 7, 12, 31, 55, 152, 273, 790, 1428, 4271, 7752, 23767, 43263, 135221,

    246675, 782968, 1430715, 4598804, 8414640, 27332956, 50067108, 164081764]



The sequence enumerating order-ideals in P_{n+1,n+2} that alternate colors, \

    with coloring parameter n mod 2, up to the , 23, -th term  is

                              starting with n=0 is


This sequence is our object of desire, the elusive sequence enumerating ODD (n+1,n+2)-core partitions 

[1,2, 4, 7, 17, 31, 80, 152, 404, 790, 2140, 4271, 11729, 23767, 65952, 135221, 
378321, 782968, 2205168, 4598804, 13023324, 27332956, 77761008]



The sequence enumerating order-ideals in P_{n+1,n+2} that alternate colors, \

    with coloring parameter n+1 mod 2, up to the , 23, -th term  is

                              starting with n=0 is


This sequence is (presumably) the twin-sequence, the good-looking A047749, defined by
If n=2m then C(3m,m)/(2m+1); if n=2m+1 then C(3m+1,m+1)/(2m+1) 

[1, 2, 3, 7, 12, 30, 55, 143, 273, 728, 1428, 3876, 7752, 21318, 43263, 120175, 
246675, 690690, 1430715, 4032015, 8414640, 23841480, 50067108]


                        This took, 142028.513, seconds