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

%I #10 Dec 27 2023 21:36:11

%S 1,0,1,2,2,13,21,67,231,509,1947,5522,16637,58030,170547,579290,

%T 1896475,6081303,20884509,68398930,231286693,788124656,2649341358,

%U 9130259705,31203913903,107304612514,372144639423,1285741209096,4480102404983,15625089552273,54591352088818,191664831925204,673088362068478

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

%H A. Bostan, K. Raschel, B. Salvy, <a href="http://dx.doi.org/10.1016/j.jcta.2013.09.005">Non-D-finite excursions in the quarter plane</a>, J. Comb. Theory A 121 (2014) 45-63, Table 1 Tag 3, Tag 6

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

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

%K nonn,walk

%O 0,4

%A _Manuel Kauers_, Nov 18 2008