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A316649
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Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.
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1
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1, 0, 1, 2, 0, 1, 7, 12, 6, 0, 1, 14, 55, 92, 70, 20, 0, 1, 23, 153, 471, 780, 720, 350, 70, 0, 1, 34, 336, 1584, 4251, 7002, 7238, 4592, 1638, 252, 0, 1, 47, 640, 4210, 16175, 39733, 65226, 72660, 54390, 26250, 7392, 924, 0, 1, 62, 1107, 9596, 49225, 164898, 380731, 623576, 732618, 614700, 360162, 140184, 32604, 3432
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OFFSET
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0,4
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LINKS
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EXAMPLE
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Triangle begins:
1;
0, 1, 2;
0, 1, 7, 12, 6;
0, 1, 14, 55, 92, 70, 20;
0, 1, 23, 153, 471, 780, 720, 350, 70;
0, 1, 34, 336, 1584, 4251, 7002, 7238, 4592, 1638, 252;
...
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MAPLE
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b:= proc(n, m) option remember; expand(`if`(n+m=0, 1, add(add(
`if`(i+j=0, 0, b(sort([n-i, m-j])[])*x), j=0..m), i=0..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
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MATHEMATICA
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Join[{{1}}, Table[a =Sort[Level[Table[Table[{i, j}, {i, 0, n}], {j, 0, n}], {2}]]; f[list1_, list2_] :=Boole[(list1 - list2)[[1]] < 1 \[And] (list1 - list2)[[2]] < 1]; m = Table[Table[f[a[[l]], a[[k]]], {k, 1, Length[a]}], {l, 1, Length[a]}]; Prepend[Table[
MatrixPower[m - IdentityMatrix[(n + 1)^2], k][[1, (n + 1)^2]], {k, 1, 2 n}], 0], {n, 1, 7}]] // Grid
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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