A316675 Triangle read by rows: T(n,k) gives the number of ways to stack n triangles in a valley so that the right wall has k triangles for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 4, 3, 4, 3, 2, 1, 1, 1, 0, 0, 1, 1, 5, 4, 5, 4, 3, 2, 1, 1, 1

Offset

0,33

Links

Seiichi Manyama, Rows n = 0..50, flattened

Formula

For m >= 0,

Sum_{n>=2m} T(n,2m) *x^n = x^(2m) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).

Sum_{n>=2m+1} T(n,2m+1)*x^n = x^(2m+1) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).

Example

T(8,4) = 3.
    *                             *
   / \                           / \
  *---*   *     *---*---*       *---*
   \ / \ / \     \ / \ / \     / \ / \
    *---*---*     *---*---*   *---*---*
     \ / \ /       \ / \ /     \ / \ /
      *---*         *---*       *---*
       \ /           \ /         \ /
        *             *           *
Triangle begins:
1;
0, 1;
0, 0, 1;
0, 0, 1, 1;
0, 0, 1, 1, 1;
0, 0, 1, 1, 1, 1;
0, 0, 1, 1, 1, 1,  1;
0, 0, 1, 1, 2, 1,  1,  1;
0, 0, 1, 1, 3, 2,  1,  1,  1;
0, 0, 1, 1, 3, 3,  2,  1,  1,  1;
0, 0, 1, 1, 3, 3,  3,  2,  1,  1,  1;
0, 0, 1, 1, 4, 3,  4,  3,  2,  1,  1, 1;
0, 0, 1, 1, 5, 4,  5,  4,  3,  2,  1, 1, 1;
0, 0, 1, 1, 5, 5,  6,  5,  4,  3,  2, 1, 1, 1;
0, 0, 1, 1, 5, 5,  8,  6,  5,  4,  3, 2, 1, 1, 1;
0, 0, 1, 1, 6, 5, 10,  8,  7,  5,  4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 7, 6, 11, 10, 10,  7,  5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 7, 7, 13, 11, 12, 10,  7, 5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 7, 7, 16, 13, 14, 12, 10, 7, 5, 4, 3, 2, 1, 1, 1;

Crossrefs

Row sums give A006950.

Sums of even columns give A059777.

Cf. A004525, A000933, A089597, A014670, A316718, A316719, A316720, A316721, A316722.

Cf. A072233.

Sequence in context: A191340 A211229 A335621 * A111405 A089053 A214979

Adjacent sequences: A316672 A316673 A316674 * A316676 A316677 A316678

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 10 2018

Status

approved