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A157513 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.

%I A157513 #18 May 27 2016 03:17:32
%S A157513 1,1,1,2,7,2,5,37,38,5,14,177,390,187,14,42,806,3065,3175,874,42,132,
%T A157513 3566,20742,37260,22254,3958,132,429,15485,127575,351821,365433,
%U A157513 141442,17548,429,1430,66373,734332,2876886,4597444,3100670,839068,76627,1430
%N A157513 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 A157513 The first and the last terms in each row are Catalan numbers. The sum in each row gives the Gessel sequence.
%H A157513 Alois P. Heinz, <a href="/A157513/b157513.txt">Table of n, a(n) for n = 0..5049</a>
%H A157513 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 A157513 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 A157513 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 A157513 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 A157513 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 A157513 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 A157513 Triangle begins:
%e A157513 1;
%e A157513 1,     1;
%e A157513 2,     7,    2;
%e A157513 5,    37,   38,    5;
%e A157513 14,  177,  390,  187,   14;
%e A157513 42,  806, 3065, 3175,  874,  42;
%p A157513 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 A157513 seq (seq (T(n, k), k=0..n), n=0..8);  # _Alois P. Heinz_, Jul 04 2011
%t A157513 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 A157513 Cf. A135404, A000531, A000108.
%K A157513 nonn,tabl,walk
%O A157513 0,4
%A A157513 _Arvind Ayyer_, Mar 02 2009