%I
%S 1,0,1,2,4,8,31,74,180,598,1834,4788,14996,49482,143789,435354,
%T 1453804,4561596,13938918,45872900,150436644,473005586,1538113866,
%U 5131580610,16631971200,54060256628,180817311524,599398780812,1965766211768,6566285301920,22047832597671,73217323048944,244741654013110
%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, 1), (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, 1 + 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 0,4
%A _Manuel Kauers_, Nov 18 2008
