editing

approved

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), (1, 0), (1, 1)}.

approved

editing

M. Bousquet-Melou 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>.

_Manuel Kauers (manuel(AT)kauers.de), _, Nov 18 2008

M. BouquetBousquet-Melou and M. Mishna, 2008. Walks with small steps in the quarter plane, <a href="http://arxiv.org/abs/0810.4387">ArXiv 0810.4387</a>.

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

nonn,walk,new

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

1, 1, 4, 13, 51, 207, 887, 3907, 17689, 81598, 382809, 1819544, 8748842, 42469534, 207900762, 1025103628, 5087012042, 25386558037, 127331796354, 641546957748, 3245566636974, 16479875939807, 83960810598237, 429073494547532, 2198921270348087, 11298292559488283, 58190917785493662, 300372279767201773

0,3

M. Bouquet-Melou and M. Mishna, 2008. Walks with small steps in the quarter plane, <a href="http://arxiv.org/abs/0810.4387">ArXiv 0810.4387</a>.

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[-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}]

nonn,walk

Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008

approved