A151378 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, 1, 2, 4, 7, 16, 37, 78, 191, 472, 1091, 2784, 7152, 17460, 45788, 120682, 305441, 816698, 2192203, 5691926, 15441378, 42019320, 111194238, 305061352, 839027467, 2253118896, 6237085427, 17301189778, 47005703891, 131077143762, 366148478721, 1004274998298, 2817566773469, 7916593883722, 21885125128391

Offset

0,4

Links

Table of n, a(n) for n=0..35.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.

Formula

G.f.: (1+x)^(1/2)*(1-3*x)^(3/2)*Int((hypergeom([-2/3,-1/3],[1],27*x^3)-1+4*x*(hypergeom([-1/3,1/3],[2],27*x^3)-1)+9*x^2)/(3*(1+x)^(3/2)*(1-3*x)^(5/2)),x)/x^3. - Mark van Hoeij, Aug 25 2014

Mathematica

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[1 + i, j, -1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A259545 A319559 A260790 * A192464 A360885 A343869

Adjacent sequences: A151375 A151376 A151377 * A151379 A151380 A151381

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved