Here's a quick breakdown of F(n,k) for n < 8. Note that F(n,n) is 0 for n<3. I haven't looked at this in depth, so I'll be staring at it as well.

1
1	1
1	1	1	1
1	2	4	5	4	2	1
1	2	7	16	26	32	26	16	7	2	1
1	3	14	50	133	280	457	580	580	457	280	133	50	14	3	1
1	3	21	109	459	1512	3972	8451	14724	21226	25446	25446	21226	14724	8451	3972	1512	459	109	21	3	1