A135404 Gessel sequence: the number of paths of length 2m in the plane, starting and ending at (0,1), with unit steps in the four directions (north, east, south, west) and staying in the region y > 0, x > -y.

1, 2, 11, 85, 782, 8004, 88044, 1020162, 12294260, 152787976, 1946310467, 25302036071, 334560525538, 4488007049900, 60955295750460, 836838395382645, 11597595644244186, 162074575606984788, 2281839419729917410, 32340239369121304038, 461109219391987625316, 6610306991283738684600

Offset: 0

Author

Doron Zeilberger, Dec 11 2007

Keywords

• nonn
• walk

Comments

An equivalent definition: number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 2 n steps taken from {(-1, -1), (-1, 0), (1, 0), (1, 1)}

According to Ira Gessel's student, Guoce Xin, Ira Gessel made his intriguing conjecture in 2001.

On June 25, 2008, the Gessel Conjecture became the Kauers-Koutschan-Zeilberger theorem - see the link. - Doron Zeilberger, Jul 01 2008

Examples

			a(1)=2 since there are only two walks, starting and ending at (0,1), of length 2, that stay in y>0, x>-y, namely: NS, EW. The other two walks, SN, WE, venture outside the allowed region.
G.f. = 1 + 2*x + 11*x^2 + 85*x^3 + 782*x^4 + 8004*x^5 + 1020162*x^6 + ...
		

References

• I. Gessel, private communication.

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
• A. Ayyer, Towards a Human Proof of Gessel's Conjecture, JIS 12 (2009) 09.4.2.
• Alin Bostan and Manuel Kauers; with an appendix by Mark van Hoeij, The complete generating function for Gessel walks is algebraic, Proc. Amer. Math. Soc. 138 (2010), 3063-3078.
• M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
• Mireille Bousquet-Mélou, An elementary solution of Gessel's walks in the quadrant, arXiv:1503.08573 [math.CO], 2015.
• Timothy Budd, Winding of simple walks on the square lattice, arXiv:1709.04042 [math.CO], 2017.
• S. Garrabrant, I. Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014.
• Manuel Kauers, Christoph Koutschan and Doron Zeilberger, Proof of Ira Gessel's Lattice Path Conjecture; Local copy [Pdf file only, no active links]
• M. Kauers and D. Zeilberger, The Quasi-Holonomic Ansatz and Restricted Lattice Walks. J. Difference Equ. Appl. 14 (2008), no. 10-11, 1119-1126; Local copy of Zeilberger's Introduction [Pdf file only, no active links]
• Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.

Crossrefs

Cf. A060900 (gives the total number of walks, regardless of final destination).

Programs

Maple

See the Maple package QuarterPlane in the webpage http://www.math.rutgers.edu/~zeilberg/tokhniot/QuarterPlane. See in particular Procedure W, which can handle any set of steps. Gessel's problem is equivalent to walks in the positive quarter-plane, starting and ending at the origin, with steps {E,W,NE,SW}.
# Maple code from N. J. A. Sloane, Mar 31 2015:
rf:=proc(a,n) mul(a+i,i=0..n-1); end;
f:=n->16^n*rf(5/6,n)*rf(1/2,n)/(rf(5/3,n)*rf(2,n));
[seq(f(n),n=0..50)];

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, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 2 n], {n, 0, 25}]
a[ n_] := If[ n<0, 0, 16^n Pochhammer[ 5/6, n] Pochhammer[ 1/2, n] / Pochhammer[ 5/3, n] / Pochhammer[2, n]] (* Michael Somos, Jun 30 2011 *)
FullSimplify[Table[(2^(2/3+4n)Gamma[4/3]Gamma[1/2+n]Gamma[5/6+n])/(Pi Gamma[5/3+n]Gamma[2+n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 10 2016 *)

Formula

The Ira Gessel Conjecture is that a(m)=16^m*(5/6)_m*(1/2)_m/ ((2)_m*(5/3)_m), where (a)_m:=a*(a+1)*...*(a+m-1).

This sequence is given by the simple recurrence: a(0) = 1; (10+11*n+3*n^2)*a(n+1) = (20+64*n+48*n^2)*a(n). - Iwan Jensen (I.Jensen(AT)ms.unimelb.edu.au), Jul 01 2008

G.f.: (1/(2*x)) * (hypergeom([ -1/2, -1/6], [2/3], 16 * x)-1). - Mark van Hoeij, Nov 02 2009

G.f.: hypergeom([1/2, 5/6, 1], [5/3, 2], 16*x). - Mark van Hoeij, Nov 02 2009

G.f.: (T(x)-1)/(2*x) where T(x) satisfies 27*T^8-18*(1+256*x^2+224*x)*T^4-8*(16*x+1)*(256*x^2-544*x+1)*T^2-(1+256*x^2+224*x)^2 = 0. - Mark van Hoeij, Nov 02 2009

G.f.: (1/(8*x)) * (27*T^7-21*T^3+(256*x-2)*T-4) where T satisfies 27*T^8-18*T^4+(-8+256*x)*T^2-1 = 0, T(0)=1. - Mark van Hoeij, Nov 02 2009

G.f.: (T(x)-1)/(2*x) where T(x) satisfies T(x(1+x)^3/(1+4x)^3) = (1+8x+4x^2)/(1+4x)^(3/2). - Ira M. Gessel, Mar 06 2013

a(n) ~ 2^(2/3)*GAMMA(1/3)/(3*Pi) * 16^n/n^(7/3). - Vaclav Kotesovec, Aug 11 2013

a(n) = (2^(2/3+4*n) * Gamma(4/3) * Gamma(1/2+n) * Gamma(5/6+n)) / (Pi*Gamma(5/3+n) * Gamma(2+n)). - Benedict W. J. Irwin, Aug 10 2016

Extensions

More terms from Manuel Kauers, Nov 18 2008

Edited by N. J. A. Sloane, Aug 28 2010