A151291 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (1, -1), (1, 0), (1, 1)}.

1, 2, 7, 23, 84, 301, 1127, 4186, 15891, 60128, 230334, 881299, 3397561, 13095693, 50725377, 196537671, 764061552, 2971863841, 11587071431, 45203638492, 176663521046, 690848304886, 2705270950914, 10599701885837, 41576025457459, 163167064273475, 640912413492991, 2518764831588455

Offset

0,2

Links

Table of n, a(n) for n=0..27.

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Formula

G.f.: Int(x*(4+Int((1-4*x)^(1/2)*(x+1/2)*(1+((1-x)*hypergeom([1/2, 3/2],[1],16*x^2/(1+4*x^2))-(1+x)*(8*x^2-4*x+1)*hypergeom([1/2, 1/2],[1],16*x^2/(1+4*x^2)))/(2*(1+4*x^2)^(1/2)*x*(2*x+1)))/x^2,x))/(1-4*x)^(3/2),x)/(x*(x-1)). - Mark van Hoeij, Aug 16 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[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A346627 A047002 A127497 * A150334 A150335 A150336

Adjacent sequences: A151288 A151289 A151290 * A151292 A151293 A151294

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved