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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. 1
1, 1, 1, 2, 7, 2, 5, 37, 38, 5, 14, 177, 390, 187, 14, 42, 806, 3065, 3175, 874, 42, 132, 3566, 20742, 37260, 22254, 3958, 132, 429, 15485, 127575, 351821, 365433, 141442, 17548, 429, 1430, 66373, 734332, 2876886, 4597444, 3100670, 839068, 76627, 1430 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The first and the last terms in each row are Catalan numbers. The sum in each row gives the Gessel sequence.
LINKS
Arvind Ayyer, Towards a human proof of Gessel's conjecture, arXiv:0902.2329 [math.CO], 2009.
Manuel Kauers, Christoph Koutschan and Doron Zeilberger, Proof of Ira Gessel's Lattice Path Conjecture
Marko Petkovsek and Herbert S. Wilf, On a conjecture of Ira Gessel, arXiv:0807.3202 [math.CO], 2008.
EXAMPLE
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)];
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)];
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)].
Triangle begins:
1;
1, 1;
2, 7, 2;
5, 37, 38, 5;
14, 177, 390, 187, 14;
42, 806, 3065, 3175, 874, 42;
MAPLE
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):
seq (seq (T(n, k), k=0..n), n=0..8); # Alois P. Heinz, Jul 04 2011
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A021371 A332501 A365524 * A087706 A102447 A151869
KEYWORD
nonn,tabl,walk
AUTHOR
Arvind Ayyer, Mar 02 2009
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)