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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)}.
0

%I #12 Dec 27 2023 21:29:18

%S 1,1,3,6,21,52,193,532,2034,5985,23283,71610,281688,894660,3545919,

%T 11541114,45988056,152599174,610459630,2057627572,8255756937,

%U 28190598072,113365682625,391366771432,1576636806694,5494187368180,22164611973120,77867785091880,314485620041700,1112719363776720,4498007052780945

%N 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)}.

%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.

%H Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, <a href="https://arxiv.org/abs/2103.04599">A decomposition of ballot permutations, pattern avoidance and Gessel walks</a>, arXiv:2103.04599 [math.CO], 2021.

%t 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}]

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008