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A322909
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The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.
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4
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1, 1, 7, 100, 2840, 129428, 8613997, 791557152, 95921167710, 14818153059968, 2842735387366627, 663020104070865664, 184757202542187563476, 60623405966739216871680, 23135486197103263598936745, 10160292704659539620791062528, 5087671168376607498331875818106
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OFFSET
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0,3
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COMMENTS
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The matrix M(n) differs from that of A306457 in using successive positive integers in place of successive prime numbers. [Modified by Stefano Spezia, Dec 20 2019 at the suggestion of Michel Marcus]
The sum of the first row of M(n) is A000217(n).
For n > 1, the sum of the superdiagonal of M(n) is A005843(n).
For n > 1 and k > 0, the sum of the k-th subdiagonal of M(n) is A120070(n,k). - Stefano Spezia, Dec 31 2019
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LINKS
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EXAMPLE
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For n = 1 the matrix M(1) is
1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
1, 2
3, 1
with permanent a(2) = 7.
For n = 3 the matrix M(3) is
1, 2, 3
4, 1, 2
5, 4, 1
with permanent a(3) = 100.
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MAPLE
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with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(ToeplitzMatrix([
seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))):
seq(a(n), n = 0 .. 15);
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MATHEMATICA
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b[n_]:=n; a[n_]:=If[n==0, 1, Permanent[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]]; Array[a, 15, 0]
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PROG
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(PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, n+i-1)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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