A060900 Number of walks of length n on square lattice, starting at origin, staying on points with x >= 0, y <= x.

1, 2, 7, 21, 78, 260, 988, 3458, 13300, 47880, 185535, 680295, 2649570, 9841260, 38470380, 144263925, 565514586, 2136388436, 8392954570, 31893227366, 125515281892, 479240167224, 1888770070824, 7240285271492, 28569774314536, 109883747363600, 434040802086220

Offset

0,2

Comments

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; https://specfun.inria.fr/bostan/HDR.pdf

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.

A. Bostan, Computer Algebra for Lattice Path Combinatorics, 7th Séminaire Lotharingien de Combinatoire, Ellwangen, March 23-25, 2015.

Formula

The following conjectural formula for this sequence is apparently due to Ira M. Gessel: a(0) = 1, a(2n) = a(2n-1)*(12n+2)/(3n+1), a(2n+1) = a(2n)*(4n+2)/(n+1).

G.f.: (hypergeom([ -1/12, 1/4],[2/3],-64*x*(4*x+1)^2/(4*x-1)^4)-1)/(2*x). - Mark van Hoeij, Nov 02 2009

G.f.: (T(x)-1)/(2*x) where T(x) satisfies 27*(4*x-1)^2*T^8 - 18*(4*x-1)^2*T^4 - (128*x^2+192*x+8)*T^2 - (4*x-1)^2 = 0. - Mark van Hoeij, Nov 02 2009

a(n) ~ 4^(n+1) / (sqrt(3) * Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Sep 17 2017

Maple

b:= proc(n, x, y) option remember;
      `if`(x<0 or y>x, 0, `if`(n=0, 1, add(add(
       b(n-1, x+i, y+j), j=[-1, 1]), i=[-1, 1])))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 30 2015

Mathematica

(* Conjectural *) a[0]=1; a[n_] := a[n] = If[EvenQ[n], (4*(3*n+1)*a[n-1])/ (3*n+2), (4*n*a[n-1])/(n+1)]; Table[a[n], {n, 0, 26}]
(* or, from 1st g.f. *) s = (HypergeometricPFQ[{-1/12, 1/4}, {2/3}, -64*x* (4*x+1)^2/(4*x-1)^4]-1)/(2*x) + O[x]^27; CoefficientList[s, x](* Jean-François Alcover, Nov 30 2015 *)

Crossrefs

Cf. A005566, A001700, A060897, A060898, A060899.

Sequence in context: A127540 A339041 A319852 * A305850 A151289 A150300

Adjacent sequences: A060897 A060898 A060899 * A060901 A060902 A060903

Keyword

nonn

Author

David W. Wilson, May 05 2001

Extensions

Entry revised by N. J. A. Sloane at the suggestion of Doron Zeilberger, Sep 13 2007

Status

approved