A122951 - OEIS

%I #23 Sep 16 2024 02:51:58

%S 1,1,5,22,117,654,3843,23323,145172,921508,5942737,38825546,256431172,

%T 1709356836,11485249995,77703736926,528893901963,3619228605738,

%U 24884558358426,171828674445330,1191050708958096,8284698825305832

%N Number of walks from (0,0) to (n,n) in the region x >= y with the steps (1,0), (0,1), (2,0) and (0,2).

%C When this walk is further restricted to the subset of the plane x-y <= 2, this gives the sequence A046717. Similarly, the sequence for such a walk restricted to x-y <= w (w > 2) is not present in the OEIS. The reference provided proves recurrences for generating functions in w.

%H Robert Israel, <a href="/A122951/b122951.txt">Table of n, a(n) for n = 0..1000</a>

%H Arvind Ayyer and Doron Zeilberger, <a href="https://arxiv.org/abs/math/0610734">The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points</a>, arXiv:math/0610734 [math.CO], 2006-2007.

%F In Maple, GF is given by solve(z^4*F^4 -2*z^3*F^3 -z^2*F^3 +2*z^2*F^2 +3*z*F^2 -2*z*F-F+1, F);

%e a(2) = 5 because we can reach (2,2) in the following ways:

%e   (0,0),(1,0),(1,1),(2,1),(2,2)

%e   (0,0),(2,0),(2,2)

%e   (0,0),(1,0),(2,0),(2,2)

%e   (0,0),(2,0),(2,1),(2,2)

%e   (0,0),(1,0),(2,0),(2,1),(2,2).

%p N:= 100: # to get a[0] to a[N]

%p S:= series(RootOf(z^4*F^4-2*z^3*F^3-z^2*F^3+2*z^2*F^2+3*z*F^2-2*z*F-F+1,F), z, N+1):

%p seq(coeff(S,z,j),j=0..N); # _Robert Israel_, Feb 18 2013

%t f[x_] = (2x+Sqrt[4(x-2)x+1] - Sqrt[2]Sqrt[2x(-2x + Sqrt[4(x-2)x+1]-1) + Sqrt[4(x-2)x+1]+1]+1)/(4x^2);

%t CoefficientList[Series[f[x],{x,0,21}],x]

%t (* _Jean-François Alcover_, May 19 2011, after g.f. *)

%Y Cf. A000108, A046717.

%K nonn,nice,walk

%O 0,3

%A _Arvind Ayyer_, Oct 25 2006