A151501 - OEIS

A151501

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), (-1, 1), (0, -1), (1, 0)}

%I #5 Dec 04 2016 13:57:04

%S 1,0,2,1,8,12,45,113,341,1018,3095,9555,30324,95537,309763,1002577,

%T 3285362,10859463,36025440,120586518,405317722,1369340161,4651457133,

%U 15854097472,54284662488,186521139750,642995941300,2224213237576,7715127092600,26838793844166,93608792582527,327272805790058,1146930142662274

%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), (-1, 1), (0, -1), (1, 0)}

%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, j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + 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