A238763 - OEIS

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

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OFFSET

0,6

COMMENTS

Similar to A020474 but with a different enumeration.

Compare with the definition of the generalized ballot numbers A238762.

LINKS

Table of n, a(n) for n=0..74.

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).

EXAMPLE

[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.

PROG

(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)]

CROSSREFS

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

Sequence in context: A179461 A364568 A364499 * A029186 A283023 A372554

Adjacent sequences: A238760 A238761 A238762 * A238764 A238765 A238766

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Mar 05 2014

STATUS

approved