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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 Alois P. Heinz, Table of n, a(n) for n = 0..5049 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 Cf. A135404, A000531, A000108. Sequence in context: A074473 A021371 A332501 * A087706 A102447 A151869 Adjacent sequences:  A157510 A157511 A157512 * A157514 A157515 A157516 KEYWORD nonn,tabl,walk AUTHOR Arvind Ayyer, Mar 02 2009 STATUS approved

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Last modified December 3 12:45 EST 2021. Contains 349463 sequences. (Running on oeis4.)