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Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.
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%I #18 May 27 2016 03:17:32

%S 1,1,1,2,7,2,5,37,38,5,14,177,390,187,14,42,806,3065,3175,874,42,132,

%T 3566,20742,37260,22254,3958,132,429,15485,127575,351821,365433,

%U 141442,17548,429,1430,66373,734332,2876886,4597444,3100670,839068,76627,1430

%N Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.

%C The first and the last terms in each row are Catalan numbers. The sum in each row gives the Gessel sequence.

%H Alois P. Heinz, <a href="/A157513/b157513.txt">Table of n, a(n) for n = 0..5049</a>

%H Arvind Ayyer, <a href="http://arxiv.org/abs/0902.2329">Towards a human proof of Gessel's conjecture</a>, arXiv:0902.2329 [math.CO], 2009.

%H Manuel Kauers, Christoph Koutschan and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gessel.html">Proof of Ira Gessel's Lattice Path Conjecture</a>

%H Marko Petkovsek and Herbert S. Wilf, <a href="http://arxiv.org/abs/0807.3202">On a conjecture of Ira Gessel</a>, arXiv:0807.3202 [math.CO], 2008.

%e For n=2, there are 2 walks of length 4 where the diagonal steps (1,1) and (-1,-1) occur zero times [(1,0),(1,0),(-1,0),(-1,0)] and [(1,0),(-1,0),(1,0),(-1,0)];

%e 7 walks where the diagonal steps occur once [(1,0),(-1,0),(1,1),(-1,-1)], [(1,1),(-1,-1),(1,0),(-1,0)], [(1,0),(1,1),(-1,0),(-1,-1)], [(1,0),(1,1),(-1,-1),(-1,0)], [(1,1),(1,0),(-1,0),(-1,-1)], [(1,1),(1,0),(-1,-1),(-1,0)], [(1,1),(-1,0),(1,0),(-1,-1)];

%e and finally 2 walks where the diagonal steps occur twice [(1,1),(1,1),(-1,-1),(-1,-1)] and [(1,1),(-1,-1),(1,1),(-1,-1)].

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 7, 2;

%e 5, 37, 38, 5;

%e 14, 177, 390, 187, 14;

%e 42, 806, 3065, 3175, 874, 42;

%p b:= proc(n, k, t, x, y) option remember; `if` (min(n, x, y, k, t, n-x)<0, 0, `if` (n=0, `if` (max(n, k, t)=0, 1, 0), b(n-1, k-1, t, x+1, y+1) +b(n-1, k, t, x+1, y) +b(n-1, k, t, x-1, y) +b(n-1, k, t-1, x-1, y-1))) end: T:= (n,k)-> b(2*n, k, k, 0, 0):

%p seq (seq (T(n, k), k=0..n), n=0..8); # _Alois P. Heinz_, Jul 04 2011

%t b[n_, k_, t_, x_, y_] := b[n, k, t, x, y] = If[Min[n, x, y, k, t, n-x] < 0, 0, If[n == 0, If[Max[n, k, t] == 0, 1, 0], b[n-1, k-1, t, x+1, y+1] + b[n - 1, k, t, x+1, y] + b[n-1, k, t, x-1, y] + b[n-1, k, t-1, x-1, y-1]]]; T[n_, k_] := b[2*n, k, k, 0, 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 27 2016, after _Alois P. Heinz_ *)

%Y Cf. A135404, A000531, A000108.

%K nonn,tabl,walk

%O 0,4

%A _Arvind Ayyer_, Mar 02 2009