%I #8 Dec 27 2023 21:29:42
%S 1,1,2,3,8,15,44,91,286,633,2072,4796,16180,38727,133548,327895,
%T 1150226,2881857,10247072,26099008,93830568,242264113,878987980,
%U 2295723288,8393889812,22139557300,81484257064,216757035756,802324361696,2150196732767,7998562017700,21576806128743,80615634684738
%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), (0, 1), (1, -1)}.
%C Apparently a duplicate of A148005. [From _R. J. Mathar_, Dec 13 2008]
%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, 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