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A306457
The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n) and whose first column consists of prime(1), prime(n + 1), ..., prime(2*n - 1).
7
1, 2, 19, 546, 40851, 4747510, 986799301, 292666754602, 135134321711681, 80312872924339660, 55242523096584443271, 52058868505260739019880, 55579419798019716586180451, 72402676504369062268839297084, 120521257466525185305708420453019, 234000358527930078723939842673115488
OFFSET
0,2
COMMENTS
The trace of the matrix M(n) is A005843(n).
The sum of the first row of the matrix M(n) is A007504(n).
The determinant of the matrix M(n) is A318173(n).
For n > 1, the subdiagonal sum of the matrix M(n) is A306192(n).
LINKS
Wikipedia, Toeplitz matrix
EXAMPLE
For n = 1 the matrix M(1) is
2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
2, 3
5, 2
with permanent a(2) = 19.
For n = 3 the matrix M(3) is
2, 3, 5
7, 2, 3
11, 7, 2
with permanent a(3) = 546.
MAPLE
f:= proc(n) uses LinearAlgebra; `if`(n=0, 1, Permanent(ToeplitzMatrix([seq(ithprime(i), i=2*n-1..n+1, -1), seq(ithprime(i), i=1..n)]))) end proc: map(f, [$0..15]);
MATHEMATICA
p[i_]:=Prime[i]; a[n_]:=If[n==0, 1, Permanent[ToeplitzMatrix[Join[{p[1]}, Array[p, n-1, {n+1, 2*n-1}]], Array[p, n]]]]; Array[a, 15, 0]
PROG
(PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(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)); \\ Michel Marcus, Mar 16 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Stefano Spezia, Feb 17 2019
EXTENSIONS
a(0) = 1 prepended by Stefano Spezia, Dec 06 2019
STATUS
approved