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%I #7 Jan 01 2024 02:44:12
%S 1,1,2,5,14,40,122,380,1217,3967,13137,44053,149321,510735,1760687,
%T 6111171,21338857,74906438,264192659,935757437,3327107090,11870683108,
%U 42486960839,152506832992,548875377127,1980241926132,7160483990345,25946243999559,94199923539220,342620238726110,1248274347651947
%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), (1, -1), (1, 1)}.
%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, -1 + j, -1 + n] + aux[-1 + 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
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