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A316649
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.
1
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
OFFSET
0,4
LINKS
EXAMPLE
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;
...
MAPLE
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)):
seq(T(n), n=0..8); # Alois P. Heinz, Jul 10 2018
MATHEMATICA
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
CROSSREFS
Columns k=0-2 give: A000007, A057427, A008865(n+1) for n>0.
Row sums give A052141.
T(n,n) gives A108628(n-1) for n>0.
T(n,2n) gives A000984.
Cf. A007318.
Sequence in context: A078341 A371762 A199459 * A065329 A352772 A108998
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Jul 09 2018
STATUS
approved