A151377 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, 0), (0, -1), (1, 1)}.

1, 0, 1, 2, 2, 8, 21, 30, 110, 304, 522, 1828, 5188, 9904, 33805, 97398, 199382, 669152, 1946558, 4187868, 13885724, 40660272, 90804738, 298319340, 877698252, 2018087328, 6581773876, 19433552840, 45742887816, 148299461600, 439078448189, 1053631252646, 3398943033446, 10085329191360, 24595733167734

Offset

0,4

Comments

Number of Motzkin n-paths in which the partial counts of flat steps do not exceed the corresponding partial counts of up steps. Bijection: substitute the steps for directions U=NE, F=S and D=W; then the rules are the same: F<=U, D<=U, and D=U at end. David Scambler, Aug 02 2012

Links

Table of n, a(n) for n=0..34.

M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.

Example

a(3)=2: Motzkin 3-paths are UFD and UDF; the paths FFF and FUD violate the condition at the first step.

Mathematica

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A137774 A167532 A208235 * A151407 A130102 A151384

Adjacent sequences: A151374 A151375 A151376 * A151378 A151379 A151380

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved