A151497 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), (1, 0)}.

1, 0, 1, 1, 2, 5, 10, 21, 56, 126, 294, 792, 1914, 4719, 12870, 32461, 82940, 228514, 593164, 1553630, 4316572, 11438076, 30509934, 85359210, 229719538, 621361560, 1748563730, 4762954770, 13025860020, 36837179385, 101317593480, 279567203745, 793968265740, 2201121030510, 6118456219260, 17440129166730

Offset

0,5

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, 2008.

Formula

G.f.: (1-2*x-3*x^2)^(1/2)*(x-1/3)*Int((27*x^3+3*x^2-x-1-(3*x-1)*hypergeom([-2/3, -1/3],[1],27*x^3)-4*x*(3*x-1)*hypergeom([-1/3, 1/3],[2],27*x^3))/((1-2*x-3*x^2)^(3/2)*(3*x-1)^2),x)/x^3. - Mark van Hoeij, Aug 20 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, 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}]

Crossrefs

Sequence in context: A192317 A361779 A360748 * A263307 A278441 A323939

Adjacent sequences: A151494 A151495 A151496 * A151498 A151499 A151500

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved