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A151331 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, 1), (-1, 0), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)}. 0
1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, 602215614, 4450146624, 33076800900, 247096919784, 1854031805769, 13965171795432, 105550935041552, 800212396412000, 6083310009164388, 46360755048406656, 354109165968099048, 2710276234371255888, 20782807250217463750 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..23.

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, 2008.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, ArXiv 0810.4387, 2008.

FORMULA

G.f.: (1/x)*Int(-(16*x^2+24*x-1)/(1+4*x)^5*hypergeom([5/4, 5/4],[2],-2*x/(x+1/4)^4*(x+1)*(x-1/8)),x). - Mark van Hoeij, Oct 13 2009

G.f.: Int(hypergeom([3/2,3/2],[2],16*x*(1+x)/(1+4*x)^2)/(1+4*x)^3,x)/x. - Mark van Hoeij, Aug 14 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[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + 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}]

CoefficientList[Series[Integrate[HypergeometricPFQ[{3/2, 3/2}, {2}, 16*x*(1+x)/(1+4*x)^2]/(1+4*x)^3, x]/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 16 2014, after Mark van Hoeij *)

CROSSREFS

Sequence in context: A106328 A007277 A025595 * A137962 A267662 A169604

Adjacent sequences:  A151328 A151329 A151330 * A151332 A151333 A151334

KEYWORD

nonn,walk

AUTHOR

Manuel Kauers, Nov 18 2008

STATUS

approved

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Last modified March 5 14:57 EST 2021. Contains 341823 sequences. (Running on oeis4.)