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%I #7 Jan 01 2024 02:33:20
%S 1,0,1,1,4,5,20,47,126,327,1041,2854,8083,24892,76208,222509,686079,
%T 2174516,6673802,20646366,66232829,210542762,662048460,2129820354,
%U 6910328426,22163635347,71624432296,234661293959,765583825870,2494320434589,8213432381547,27116116390705,89225059889125,295111869092137
%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), (-1, 1), (0, -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[i, 1 + j, -1 + n] + aux[1 + 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 0,5
%A _Manuel Kauers_, Nov 18 2008