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%I #8 Dec 27 2023 21:34:33
%S 1,0,2,1,10,14,75,178,738,2304,8753,31186,117244,444107,1698959,
%T 6637212,25978768,103470800,413378548,1671251159,6788681873,
%U 27799009080,114434701571,473862635406,1972068523477,8246133103745,34634859230481,146059305983882,618344086849853,2627035342133955
%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, 0), (-1, 1), (0, -1), (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 23.
%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[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008