This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382776 #13 Apr 05 2025 12:01:44 %S A382776 1,1,1,6,9,6,90,180,180,90,2520,6300,8100,6300,2520,113400,340200, %T A382776 529200,529200,340200,113400,7484400,26195400,47628000,57153600, %U A382776 47628000,26195400,7484400,681080400,2724321600,5658206400,7858620000,7858620000,5658206400,2724321600,681080400 %N 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. %C A382776 The configurations are such that k columns will each contain 2 rooks and n-k rows will each contain 2 rooks. %F A382776 T(n,k) = binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n). %F A382776 T(n,n-k) = T(n,k). %e A382776 Triangle begins: %e A382776 1; %e A382776 1, 1; %e A382776 6, 9, 6; %e A382776 90, 180, 180, 90; %e A382776 2520, 6300, 8100, 6300, 2520; %e A382776 113400, 340200, 529200, 529200, 340200, 113400; %e A382776 7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400; %e A382776 ... %e A382776 The T(2,0) = 6 configurations are: %e A382776 X X . . X . X . X . . X . X X . . X . X . . X X %e A382776 . . X X . X . X . X X . X . . X X . X . X X . . %e A382776 The T(2,1) = 9 configurations are: %e A382776 X X . X . X . X X . . X . X . X . . . . X . X . X . . %e A382776 . . X . X . X . . X X . X . X . X X . . X . X . X . . %e A382776 . . X . X . X . . . . X . X . X . . X X . X . X . X X %o A382776 (PARI) T(n,k)=binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n) %Y A382776 Row sums are A382777. %Y A382776 Column k=0 is A000680. %K A382776 nonn,tabl %O A382776 0,4 %A A382776 _Andrew Howroyd_, Apr 04 2025