A270227 Array read by antidiagonals: T(n,m) is the number of matchings in the rook graph K_n X K_m.

1, 2, 2, 4, 7, 4, 10, 32, 32, 10, 26, 193, 370, 193, 26, 76, 1382, 5950, 5950, 1382, 76, 232, 11719, 122984, 270529, 122984, 11719, 232, 764, 112604, 3175696, 16873930, 16873930, 3175696, 112604, 764, 2620, 1221889, 98815588, 1384880065, 3337807996, 1384880065, 98815588, 1221889, 2620

Offset: 1

Author

Andrew Howroyd, Mar 13 2016

Keywords

• nonn
• tabl

Comments

Observations: (for n+m <= 32)

Examination of values modulus a small prime yields several patterns.

T(n,m) == (n+1)*(m+1) (mod 2) for n+m>2.

T(n,m) == T(n,m+6) (mod 3).

T(n,m) is not divisible by 3.

T(n,m) == 0 (mod 5) for n==4 (mod 5) and m<>2 and except when m=n=4.

T(5,m) == 0 (mod 208) for m >= 13.

T(6,m) == 0 (mod 19) for m >= 19.

Examples

			The start of the sequence as table:
*   1     2       4         10           26              76 ...
*   2     7      32        193         1382           11719 ...
*   4    32     370       5950       122984         3175696 ...
*  10   193    5950     270529     16873930      1384880065 ...
*  26  1382  122984  168739305   3337807996    909046586596 ...
*  76 11719 3175696 1384880065 909046586596 855404716021831 ...
* ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..496
• Eric Weisstein's World of Mathematics, Independent Edge Set
• Eric Weisstein's World of Mathematics, Matching
• Eric Weisstein's World of Mathematics, Rook Graph

Crossrefs

Main diagonal is A270228. Rows include A000085, A270229.