A316649 Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.

1, 0, 1, 2, 0, 1, 7, 12, 6, 0, 1, 14, 55, 92, 70, 20, 0, 1, 23, 153, 471, 780, 720, 350, 70, 0, 1, 34, 336, 1584, 4251, 7002, 7238, 4592, 1638, 252, 0, 1, 47, 640, 4210, 16175, 39733, 65226, 72660, 54390, 26250, 7392, 924, 0, 1, 62, 1107, 9596, 49225, 164898, 380731, 623576, 732618, 614700, 360162, 140184, 32604, 3432

Offset: 0

Author

Geoffrey Critzer, Jul 09 2018

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  0, 1,  2;
  0, 1,  7,  12,    6;
  0, 1, 14,  55,   92,   70,   20;
  0, 1, 23, 153,  471,  780,  720,  350,   70;
  0, 1, 34, 336, 1584, 4251, 7002, 7238, 4592, 1638, 252;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened

Crossrefs

Columns k=0-2 give: A000007, A057427, A008865(n+1) for n>0.

Row sums give A052141.

T(n,n) gives A108628(n-1) for n>0.

T(n,2n) gives A000984.

Cf. A007318.

Programs

Maple

b:= proc(n, m) option remember; expand(`if`(n+m=0, 1, add(add(
     `if`(i+j=0, 0, b(sort([n-i, m-j])[])*x), j=0..m), i=0..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Jul 10 2018

Mathematica

Join[{{1}},Table[a =Sort[Level[Table[Table[{i, j}, {i, 0, n}], {j, 0, n}], {2}]];f[list1_, list2_] :=Boole[(list1 - list2)[[1]] < 1 \[And] (list1 - list2)[[2]] < 1];m = Table[Table[f[a[[l]], a[[k]]], {k, 1, Length[a]}], {l, 1, Length[a]}];Prepend[Table[
     MatrixPower[m - IdentityMatrix[(n + 1)^2], k][[1, (n + 1)^2]], {k, 1, 2 n}], 0], {n, 1, 7}]] // Grid