A316675 - OEIS

%I #51 Jul 12 2018 02:54:45

%S 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,

%T 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,

%U 1,1,4,3,4,3,2,1,1,1,0,0,1,1,5,4,5,4,3,2,1,1,1

%N 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.

%H Seiichi Manyama, <a href="/A316675/b316675.txt">Rows n = 0..50, flattened</a>

%F For m >= 0,

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

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

%e T(8,4) = 3.

%e     *                             *

%e    / \                           / \

%e   *---*   *     *---*---*       *---*

%e    \ / \ / \     \ / \ / \     / \ / \

%e     *---*---*     *---*---*   *---*---*

%e      \ / \ /       \ / \ /     \ / \ /

%e       *---*         *---*       *---*

%e        \ /           \ /         \ /

%e         *             *           *

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 0, 1;

%e 0, 0, 1, 1;

%e 0, 0, 1, 1, 1;

%e 0, 0, 1, 1, 1, 1;

%e 0, 0, 1, 1, 1, 1,  1;

%e 0, 0, 1, 1, 2, 1,  1,  1;

%e 0, 0, 1, 1, 3, 2,  1,  1,  1;

%e 0, 0, 1, 1, 3, 3,  2,  1,  1,  1;

%e 0, 0, 1, 1, 3, 3,  3,  2,  1,  1,  1;

%e 0, 0, 1, 1, 4, 3,  4,  3,  2,  1,  1, 1;

%e 0, 0, 1, 1, 5, 4,  5,  4,  3,  2,  1, 1, 1;

%e 0, 0, 1, 1, 5, 5,  6,  5,  4,  3,  2, 1, 1, 1;

%e 0, 0, 1, 1, 5, 5,  8,  6,  5,  4,  3, 2, 1, 1, 1;

%e 0, 0, 1, 1, 6, 5, 10,  8,  7,  5,  4, 3, 2, 1, 1, 1;

%e 0, 0, 1, 1, 7, 6, 11, 10, 10,  7,  5, 4, 3, 2, 1, 1, 1;

%e 0, 0, 1, 1, 7, 7, 13, 11, 12, 10,  7, 5, 4, 3, 2, 1, 1, 1;

%e 0, 0, 1, 1, 7, 7, 16, 13, 14, 12, 10, 7, 5, 4, 3, 2, 1, 1, 1;

%Y Row sums give A006950.

%Y Sums of even columns give A059777.

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

%Y Cf. A072233.

%K nonn,tabl

%O 0,33

%A _Seiichi Manyama_, Jul 10 2018