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%I #7 Jan 01 2024 02:44:23
%S 1,1,2,5,13,37,109,334,1049,3367,10992,36399,121982,412969,1410446,
%T 4854123,16818222,58617286,205384162,723043647,2556317905,9072825097,
%U 32314370528,115462754160,413775789826,1486835229283,5356049103946,19338799943334,69975443521895,253704568761743,921550558422286,3353248008103356
%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, 0)}.
%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[-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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