A151297 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (0, -1), (0, 1), (1, -1), (1, 0)}.

1, 2, 7, 26, 105, 444, 1944, 8728, 39999, 186266, 879108, 4195762, 20217136, 98220992, 480623748, 2366735352, 11720044199, 58329063714, 291607308864, 1463803013168, 7375252291592, 37285645521840, 189084169866584, 961635234827346, 4903573489214352, 25065602340246704, 128419743299166764

Offset

0,2

Links

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

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.: (1/128)*(1-62*x)/x^2+(2*x+1)/x^2*(-1/128+Int(x/((2*x+1)*(1-6*x))^(3/2)*(-3-1/2*Int(((1-6*x)/(2*x+1)/(1-8*x^2)^3)^(1/2)*((24*x^3+4*x^2+4*x+1)/x^2*hypergeom( [1/4,3/4],[1],64/(8*x^2-1)^2*(2*x+1)*x^3)+12*x*(8*x^2+4*x+1)*(1+14*x+16*x^2)/(1-8*x^2)^2*hypergeom([5/4, 7/4],[2],64/(8*x^2-1)^2*(2*x+1)*x^3)),x)),x)). [Mark van Hoeij, Oct 13 2009]

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

Crossrefs

Sequence in context: A150553 A150554 A150555 * A052706 A150556 A150557

Adjacent sequences: A151294 A151295 A151296 * A151298 A151299 A151300

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved