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A157513
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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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1
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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; internal format)
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OFFSET
| 0,4
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COMMENTS
| The first and the last terms in each row are Catalan numbers. The sum in each row gives the Gessel sequence.
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REFERENCES
| Manuel Kauers, Christoph Koutschan and Doron Zeilberger, Proof of Ira Gessel's Lattice Path Conjecture.
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LINKS
| Arvind Ayyer, Towards a human proof of Gessel's conjecture.
Marko Petkovsek and Herbert S. Wilf, On a conjecture of Ira Gessel.
Alois P. Heinz, Table of n, a(n) for n = 0..5049
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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;
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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
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CROSSREFS
| Cf. A135404, A000531, A000108.
Sequence in context: A197047 A074473 A021371 * A087706 A102447 A151869
Adjacent sequences: A157510 A157511 A157512 * A157514 A157515 A157516
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KEYWORD
| nonn,tabl,walk
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AUTHOR
| Arvind Ayyer (arvind.ayyer(AT)cea.fr), Mar 02 2009
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