%I
%S 1,1,3,7,21,63,198,648,2148,7303,25159,87771,309885,1103217,3961285,
%T 14322270,52098790,190586734,700547511,2586505466,9587737311,
%U 35667671240,133128177959,498385063314,1870940572326,7041423384111,26563151794983,100425970551436,380444157541490,1443958508666422
%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, 0)}.
%H M. BousquetMÃ©lou 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[i, 1 + j, 1 + n] + aux[1 + i, 1 + j, 1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]
%Y Row sums of A199915.
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008
