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%I #7 Dec 28 2023 23:19:56
%S 1,1,2,5,12,34,97,287,887,2761,8833,28635,93897,312257,1046368,
%T 3539825,12066283,41387197,142887038,495828190,1729130445,6057385620,
%U 21303526484,75209346309,266413953357,946691896721,3373935569795,12056675023892,43193395482458,155104308523072,558187652361721,2012940278579932
%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, 0), (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, 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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