A060899 Number of walks of length n on square lattice, starting at origin, staying on points with x+y >= 0.
1, 2, 8, 24, 96, 320, 1280, 4480, 17920, 64512, 258048, 946176, 3784704, 14057472, 56229888, 210862080, 843448320, 3186360320, 12745441280, 48432676864, 193730707456, 739699064832, 2958796259328, 11342052327424
Offset: 0
Links
- Harry J. Smith, Table of n, a(n) for n=0..200
- Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
- Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
- Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
- Math Overflow, Combinatorial proof for the number of lattice paths that return to the axis only at times that are a multiple of 4, 2012.
Programs
-
Mathematica
Table[2^n Binomial[n,Floor[n/2]],{n,0,30}] (* Harvey P. Dale, Oct 15 2017 *) -
PARI
{ for (n=0, 200, write("b060899.txt", n, " ", 2^n*binomial(n, n\2)); ) } \\ Harry J. Smith, Jul 14 2009
Formula
a(n) = 2^n*binomial(n, [n/2]);
G.f.: (sqrt((1+4*x)/(1-4*x))-1)/4/x. - Vladeta Jovovic, Apr 28 2003
E.g.f.: BesselI(0, 4*x)+BesselI(1, 4*x). - Vladeta Jovovic, Apr 28 2003
a(n) = 4^n*sum{k=0..n, C(n,k)C(k)/(-2)^k}, with C(n)=A000108(n). - Paul Barry, Dec 28 2006
(n+1)*a(n) -4*a(n-1) +16*(-n+1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
a(n) = (-4)^n*hypergeom([3/2,-n],[2],2). - Peter Luschny, Apr 26 2016
Sum_{n>=0} a(n)/6^n = 3/phi = A134973. - Peter McNair, Apr 30 2022
In general, for k>4, Sum_{n>=0} a(n)/k^n = (sqrt((k+4)/(k-4)) - 1) * k/4. - Vaclav Kotesovec, May 13 2022
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=0} 1/a(n) = 16*asin(1/4)/(3*sqrt(15)) + 4/3.
Sum_{n>=0} (-1)^n/a(n) = 4/5 - 16*asin(1/4)/(5*sqrt(15)). (End)
Comments