A151396 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, -1), (0, -1), (0, 1), (1, 1)}.

1, 1, 3, 6, 21, 52, 193, 532, 2034, 5985, 23283, 71610, 281688, 894660, 3545919, 11541114, 45988056, 152599174, 610459630, 2057627572, 8255756937, 28190598072, 113365682625, 391366771432, 1576636806694, 5494187368180, 22164611973120, 77867785091880, 314485620041700, 1112719363776720, 4498007052780945

Offset

0,3

Links

Table of n, a(n) for n=0..30.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.

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[i, 1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A063683 A200380 A354017 * A148604 A148605 A148606

Adjacent sequences: A151393 A151394 A151395 * A151397 A151398 A151399

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved