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
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