A151326 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)}.

1, 3, 15, 74, 392, 2116, 11652, 64967, 365759, 2074574, 11836868, 67863126, 390625864, 2256008404, 13066434500, 75864388248, 441412162944, 2573133492918, 15024422196084, 87856077334712, 514419919265976, 3015635977208784, 17697278566338720, 103958103858046662, 611220388506542904

Offset

0,2

Links

Table of n, a(n) for n=0..24.

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)*(6*x+1)*(6+Int(3*(1-4*x-12*x^2)^(3/2)*((8*x^2-1)*(96*x^3-248*x^2-150*x-13)*hypergeom([5/4, 7/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2)+5*(8*x^2+4*x+1)*(32*x^3-32*x^2-42*x-5)*hypergeom([7/4, 9/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2))/(2*(2*x+1)*(6*x+1)^2*(1-8*x^2)^(7/2)),x))/(1-4*x-12*x^2)^(5/2),x),x)/x. - Mark van Hoeij, Aug 16 2014

a(n) ~ [x^n]( (1-2*x)*(1+2*x)^(1/2)/(4*x*(1-6*x)^(1/2)) ), where [x^n](f) denotes the coefficient of x^n in the series expansion of f. - Mark van Hoeij, May 28 2020

a(n) ~ 2^(n+1) * 3^(n - 1/2) / sqrt(Pi*n) [Bostan and Kauers, p.13]. - Vaclav Kotesovec, May 29 2020

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

Crossrefs

Sequence in context: A357222 A224397 A190010 * A063000 A002902 A236579

Adjacent sequences: A151323 A151324 A151325 * A151327 A151328 A151329

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved