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

1, 1, 4, 11, 44, 153, 638, 2443, 10468, 42306, 184576, 772024, 3411966, 14612580, 65212028, 284126776, 1277710146, 5639805613, 25519441118, 113792116639, 517542778146, 2326572672034, 10627523024268, 48093423017929, 220503392480274, 1003378635817366, 4615271568477192, 21099242329057240

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