A382776 Triangle read by rows: T(n,k) is the number of ways to place 2*n rooks on a (n+k) X (2*n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

1, 1, 1, 6, 9, 6, 90, 180, 180, 90, 2520, 6300, 8100, 6300, 2520, 113400, 340200, 529200, 529200, 340200, 113400, 7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400, 681080400, 2724321600, 5658206400, 7858620000, 7858620000, 5658206400, 2724321600, 681080400

Offset

0,4

Comments

The configurations are such that k columns will each contain 2 rooks and n-k rows will each contain 2 rooks.

Links

Table of n, a(n) for n=0..35.

Formula

T(n,k) = binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n).

T(n,n-k) = T(n,k).

Example

Triangle begins:
        1;
        1,        1;
        6,        9,        6;
       90,      180,      180,       90;
     2520,     6300,     8100,     6300,     2520;
   113400,   340200,   529200,   529200,   340200,   113400;
  7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400;
  ...
The T(2,0) = 6 configurations are:
  X X . .    X . X .    X . . X    . X X .    . X . X    . . X X
  . . X X    . X . X    . X X .    X . . X    X . X .    X X . .
The T(2,1) = 9 configurations are:
  X X .   X . X   . X X   . . X   . X .   X . .   . . X   . X .   X . .
  . . X   . X .   X . .   X X .   X . X   . X X   . . X   . X .   X . .
  . . X   . X .   X . .   . . X   . X .   X . .   X X .   X . X   . X X

Prog

(PARI) T(n, k)=binomial(2*n-k, k)*binomial(n+k, n-k)*(2*(n-k))!*(2*k)!/(2^n)

Crossrefs

Row sums are A382777.

Column k=0 is A000680.

Sequence in context: A023410 A010725 A193594 * A011480 A283743 A339764

Adjacent sequences: A382773 A382774 A382775 * A382777 A382778 A382779

Keyword

nonn,tabl

Author

Andrew Howroyd, Apr 04 2025

Status

approved