A151275 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), (1, -1), (1, 1)}.

1, 1, 5, 13, 61, 199, 939, 3389, 16129, 61601, 295373, 1169939, 5643023, 22931063, 111114587, 460242509, 2238229577, 9409963861, 45894616065, 195273012089, 954636984885, 4102002557665, 20092702499037, 87052952678691, 427107841926791, 1863539746958059, 9155858788887359, 40191781253867359

Offset

0,3

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(-1+Int((1+x)*(3*x+1)*Int(12*(1-2*x-15*x^2)^(3/2)*((8*x^2-1)*(2280*x^5+621*x^4-104*x^3-117*x^2-14*x+2)*hypergeom([7/4, 9/4],[2],64*(x^2+1)*x^2/(16*x^2+1)^2)-x*(280*x^6-297*x^5-562*x^4-406*x^3+128*x^2+11*x+10)*hypergeom([7/4, 9/4],[3],64*(x^2+1)*x^2/(16*x^2+1)^2))/((3*x+1)*(16*x^2+1)^(9/2)*(1+x)^2),x)/(1-2*x-15*x^2)^(5/2),x),x)/((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, 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: A067756 A284035 A051859 * A149567 A149568 A201805

Adjacent sequences: A151272 A151273 A151274 * A151276 A151277 A151278

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved