A057648 Number of excursions of length n on the upper-right part of the hexagonal lattice.
1, 0, 2, 2, 13, 34, 158, 594, 2665, 11558, 53320, 247488, 1181266, 5708884, 28049474, 139417402, 701063005, 3559326294, 18233244530, 94140532624, 489573775236, 2562613997512, 13493827469116, 71441865994904
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..1000
- C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001.
Programs
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Maple
gf:=(1-2*x)*hypergeom([-1/2, 1/2],[2],16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3],[2],27*x^2*(2*x+1))+1/2)/(6*x^2): S:= series(gf,x,103): seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 08 2014
Formula
G.f.: (1-2*x)*hypergeom([-1/2, 1/2],[2],16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3],[2],27*x^2*(2*x+1))+1/2)/(6*x^2). - Mark van Hoeij, Dec 08 2014
a(n) ~ (2*sqrt(3) - 3) * 2^n * 3^(n+2) / (Pi*n^3). - Vaclav Kotesovec, Apr 30 2024
Extensions
Title corrected by Sean A. Irvine, Jun 22 2022
Comments