A151329 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), (1, -1), (1, 0), (1, 1)}.

1, 3, 16, 86, 509, 3065, 19088, 120401, 771758, 4991255, 32580974, 214030883, 1414515215, 9392000723, 62625868280, 419056150250, 2812961273801, 18933686864261, 127752378798614, 863851325189957, 5852685174009545, 39721858765234379, 270018313988209400, 1838154015475958213

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 [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(1+Int((2*x+1)*(5*x+1)*(10+Int(12*(1-2*x-35*x^2)^(3/2)*((1+12*x^2)*(736*x^5+2208*x^4+1096*x^3-44*x^2+44*x+1)*hypergeom([7/4, 9/4],[2],64*(x^2+x+1)*x^2/(1+12*x^2)^2)-7*x*(1824*x^6+2496*x^5+1288*x^4+452*x^3+420*x^2+53*x+10)*hypergeom([9/4, 11/4],[3],64*(x^2+x+1)*x^2/(1+12*x^2)^2))/((5*x+1)*(1+12*x^2)^(9/2)*(2*x+1)^2),x))/(1-2*x-35*x^2)^(5/2),x),x)/((2*x+1)*x). - 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[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}]

Crossrefs

Sequence in context: A278681 A224869 A163604 * A356402 A378334 A347930

Adjacent sequences: A151326 A151327 A151328 * A151330 A151331 A151332

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved