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A199915
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Triangle read by rows: T(n,k) is the number of n-step paths from (0,0) to (0,k) that stay in the first quadrant (but may touch the axes) consisting of steps (1,0), (0,1), (0,-1) and (-1,1).
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5
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1, 0, 1, 1, 1, 1, 1, 2, 3, 1, 2, 7, 5, 6, 1, 7, 10, 21, 14, 10, 1, 10, 38, 48, 51, 35, 15, 1, 38, 89, 135, 168, 120, 76, 21, 1, 89, 229, 441, 458, 474, 281, 147, 28, 1, 229, 752, 1121, 1604, 1475, 1188, 637, 260, 36, 1, 752, 1873, 3692, 4772, 5100, 4329, 2800, 1366, 429, 45, 1
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OFFSET
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0,8
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LINKS
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EXAMPLE
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T(4,2) = 5: ((1,0),(1,0),(-1,1),(-1,1)); ((1,0),(-1,1),(1,0),(-1,1)); ((0,1),(0,1),(0,1),(0,-1)); ((0,1),(0,1),(0,-1),(0,1)); ((0,1),(0,-1),(0,1),(0,1)).
Triangle begins:
1;
0, 1;
1, 1, 1;
1, 2, 3, 1;
2, 7, 5, 6, 1;
7, 10, 21, 14, 10, 1;
10, 38, 48, 51, 35, 15, 1;
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MAPLE
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b:= proc(n, k, x, y) option remember;
`if`(n<0 or x<0 or y<0 or n<x or n<abs(k-y), 0,
`if`(n=0, 1, add (b(n-1, k, x+d[1], y+d[2]),
d=[[1, 0], [0, 1], [0, -1], [-1, 1]])))
end:
T:= (n, k)-> b(n, k, 0, 0):
seq(seq(T(n, k), k=0..n), n=0..12);
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MATHEMATICA
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b[n_, k_, x_, y_] := b[n, k, x, y] = If[n<0 || x<0 || y<0 || n<x || n<Abs[k-y], 0, If[n == 0, 1, Sum[b[n-1, k, x+d[[1]], y+d[[2]]], {d, {{1, 0}, {0, 1}, {0, -1}, {-1, 1}}}]]]; T[n_, k_] := b[n, k, 0, 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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