A151302 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), (0, 1), (1, -1), (1, 1)}.

1, 2, 8, 29, 129, 535, 2467, 10844, 50982, 231404, 1101030, 5099358, 24460460, 114857406, 554172828, 2628163725, 12736564173, 60855656041, 295934172553, 1422190689641, 6935071262465, 33482108503613, 163639829020629, 793000159238839, 3883024909162411, 18875285298740051, 92573312042858471

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.: Int(1+Int((1+x)*(3*x+1)*(6+Int(6*(1-2*x-15*x^2)^(3/2)*(4*x*(360*x^6+397*x^5+504*x^4-76*x^3+89*x^2+7*x+5)*hypergeom([7/4, 9/4],[3],64*(x^2+1)*x^2/(16*x^2+1)^2)-(8*x^2-1)*(4320*x^5+1356*x^4+112*x^3-183*x^2-38*x+1)*hypergeom([7/4, 9/4],[2],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)/((1+x)*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[i, -1 + j, -1 + n] + aux[1 + i, -1 + 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: A150752 A150753 A162066 * A150754 A150755 A150756

Adjacent sequences: A151299 A151300 A151301 * A151303 A151304 A151305

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved