A238763 A Motzkin triangle read by rows, 0<=k<=n.

1, 0, 1, 1, 0, 2, 0, 2, 0, 4, 1, 0, 5, 0, 9, 0, 3, 0, 12, 0, 21, 1, 0, 9, 0, 30, 0, 51, 0, 4, 0, 25, 0, 76, 0, 127, 1, 0, 14, 0, 69, 0, 196, 0, 323, 0, 5, 0, 44, 0, 189, 0, 512, 0, 835, 1, 0, 20, 0, 133, 0, 518, 0, 1353, 0, 2188, 0, 6, 0, 70, 0, 392, 0, 1422, 0

Offset: 0

Author

Peter Luschny, Mar 05 2014

Keywords

• nonn
• tabl

Comments

Similar to A020474 but with a different enumeration.

Compare with the definition of the generalized ballot numbers A238762.

Examples

			[n\k 0  1  2   3  4   5   6   7]
[0]  1,
[1]  0, 1,
[2]  1, 0, 2,
[3]  0, 2, 0, 4,
[4]  1, 0, 5, 0, 9,
[5]  0, 3, 0, 12, 0, 21,
[6]  1, 0, 9, 0, 30, 0, 51,
[7]  0, 4, 0, 25, 0, 76, 0, 127.
		

Crossrefs

Cf. A001006, A005043, A064189, A020474, A238762.

Programs

Sage

@CachedFunction
def T(p, q):
    if p == 0 and q == 0: return 1
    if p < 0 or  p > q: return 0
    return T(p-2, q) + T(p-1, q-1) + T(p, q-2)
[[T(p, q) for p in (0..q)] for q in (0..9)]

Formula

Definition: T(0, 0) = 1; T(p, q) = 0 if p < 0 or p > q; T(p, q) = T(p-2, q) + T(p-1, q-1) + T(p, q-2). (The notation is in the style of Knuth, TAOCP 4a (7.2.1.6)).

T(n, n) = A001006(n).

Sum_{0<=k<=n} T(n, k) = A005043(n+2).