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%I #9 Dec 27 2023 21:27:49
%S 1,1,2,4,9,23,58,160,447,1280,3799,11329,34648,107194,335052,1062809,
%T 3392928,10944072,35576811,116369798,383517144,1270497711,4232708951,
%U 14174190320,47670133454,161055621043,546226400383,1859448826545,6352403850949,21770237036258,74841611960045,258025290079584,891972398224757
%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), (0, 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[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}]
%Y Row sums of A306814.
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008