%I
%S 1,0,2,3,10,27,89,267,868,2858,9510,31830,108638,373219,1288064,
%T 4482534,15710368,55258931,195240700,693284513,2470263132,8827776270,
%U 31654190580,113835950410,410335021648,1482638356348,5369592056146,19484896375080,70835100481126,257981797359638,941120763934455,3438343699103345
%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), (1, 0), (1, 1)}
%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, 1 + j, 1 + n] + aux[1 + i, 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
