OFFSET
0,3
COMMENTS
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 trace of M(n) is A000027(n).
The sum of the first row of M(n) is A000217(n).
The sum of the first column of M(n) is A005448(n). [Corrected by Stefano Spezia, Dec 19 2019]
For n > 1, the sum of the superdiagonal of M(n) is A005843(n).
For n > 0, the sum of the (k-1)-th superdiagonal of M(n) is A003991(n,k). - Stefano Spezia, Dec 29 2019
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
LINKS
Stefano Spezia, Table of n, a(n) for n = 0..35
Wikipedia, Toeplitz Matrix
EXAMPLE
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.
MAPLE
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);
MATHEMATICA
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]
PROG
(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; }
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 19 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Stefano Spezia, Dec 30 2018
EXTENSIONS
a(0) = 1 prepended by Stefano Spezia, Dec 19 2019
STATUS
approved