# placing non-attacking knights on nXn board

Max maxale at gmail.com
Wed Apr 12 00:00:10 CEST 2006

```Hello!

There is an interesting problem http://www.mathlinks.ro/Forum/topic-80007.html

I have computed f(n) for n=1..7 and it seems to be missed in OEIS:
2, 16, 94, 1365, 55213, 3368146, 394631712

This sequence represents row sums of the following table where m-th
element (m=0,1,...) in n-th row (n=1,2,...) is the number of ways to
place m non-attacking knights on the board of size nXn.

1 1
1 4 6 4 1
1 9 28 36 18 2
1 16 96 276 412 340 170 48 6
1 25 252 1360 4436 9386 13384 12996 8526 3679 994 158 15 1
1 36 550 4752 26133 97580 257318 491140 688946 716804 556274 323476
141969 47684 12488 2560 393 40 2
1 49 1056 13340 111066 649476 2774728 8891854 21773953 41310504
61412928 72304650 68233368 52400729 33376132 18001362 8363763 3374053
1178064 351592 87913 17910 2858 336 26 1

Note that the maximum number of non-attacking knights is given by A030978.
The number of ways to place A030978(n) knights on nXn boards (the
"diagonal" in the table above) is
1,1,2,6,1,2,1
It also seems to be missing in OEIS (there are some hits but they look
unrelated).

Would someone like to verify/extend/submit these sequences?

Thanks,
Max

```