A151478 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)}.

1, 1, 4, 12, 54, 210, 1020, 4445, 22610, 105210, 551376, 2678676, 14332164, 71788860, 389991888, 1998530820, 10984120290, 57293297490, 317798892840, 1681213457352, 9395215622364, 50278804820244, 282711201927336, 1527524202392370, 8633634028624332, 47028406025950300, 266981514041485600

Offset

0,3

Links

Table of n, a(n) for n=0..26.

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

Formula

G.f.: Int(Int(((3-6*x)*hypergeom([1/2,3/2],[1],16*x/(12*x^2+8*x+1))+(6*x-1)*hypergeom([1/2,3/2],[2],16*x/(12*x^2+8*x+1)))/(12*x^2+8*x+1)^(3/2),x),x)/x^2. - Mark van Hoeij, Aug 25 2014

a(n) = A001006(n) * binomial(n,floor(n/2)). - Benedict W. J. Irwin, Oct 14 2016

Maple

seq(binomial(n, floor(n/2))*add(n!/((n-2*k)!*k!*(k+1)!), k=0..floor(n/2)), n=0..26);   # Mark van Hoeij, May 12 2013

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[-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}]

Crossrefs

Cf. A001006 (Motzkin numbers).

Sequence in context: A149415 A149416 A149417 * A125528 A124005 A149418

Adjacent sequences: A151475 A151476 A151477 * A151479 A151480 A151481

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved