%I #7 Dec 28 2023 23:20:06
%S 1,0,2,2,8,21,56,180,537,1642,5428,16894,56073,184644,608722,2060761,
%T 6912591,23511436,80469490,275394728,951992172,3293896039,11446970617,
%U 39969401778,139770090303,490850856487,1727806323657,6096204817107,21572269339099,76472370379535,271723970484170,967416967381241
%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, 0), (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, 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