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A090628
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Square array T(n,k) (row n, column k) read by antidiagonals defined by: T(n,k) is the permanent of the n X n matrix with 1 on the diagonal and k elsewhere; T(0,k)=1.
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1
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 6, 1, 1, 1, 10, 29, 24, 1, 1, 1, 17, 82, 233, 120, 1, 1, 1, 26, 177, 1000, 2329, 720, 1, 1, 1, 37, 326, 2913, 14968, 27949, 5040, 1, 1, 1, 50, 541, 6776, 58017, 269488, 391285, 40320, 1, 1, 1, 65, 834, 13609, 168376, 1393137, 5659120, 6260561, 362880, 1
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OFFSET
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0,9
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j).
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EXAMPLE
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Row n=0: 1, 1, 1, 1, 1, 1, 1, 1, ...
Row n=1: 1, 1, 1, 1, 1, 1, 1, 1, ...
Row n=2: 1, 2, 5, 10, 17, 26, 37, 50, ...
Row n=3: 1, 6, 29, 82, 177, 326, 541, 834, ...
Row n=4: 1, 24, 233, 1000, 2913, 6776, 13609, 24648, ...
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MAPLE
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T:= (n, k)-> `if`(n=0, 1, LinearAlgebra[Permanent](
Matrix(n, (i, j)-> `if`(i=j, 1, k)))):
# second Maple program:
b:= proc(n, k) b(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
(b(n-1, 0)+b(n-2, 0))), binomial(n, k)*b(n-k, 0))
end:
T:= proc(n, k) T(n, k):= add(b(n, j)*k^(n-j), j=0..n) end:
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MATHEMATICA
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T[0, _] = 1;
T[n_, k_] := Permanent[Table[If[i == j, 1, k], {i, n}, {j, n}]];
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PROG
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(PARI) T(n, k) = matpermanent(matrix(n, n, i, j, if (i==j, 1, k)));
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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