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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).

%I A306457 #18 Dec 06 2019 07:09:30
%S A306457 1,2,19,546,40851,4747510,986799301,292666754602,135134321711681,
%T A306457 80312872924339660,55242523096584443271,52058868505260739019880,
%U A306457 55579419798019716586180451,72402676504369062268839297084,120521257466525185305708420453019,234000358527930078723939842673115488
%N 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).
%C A306457 The trace of the matrix M(n) is A005843(n).
%C A306457 The sum of the first row of the matrix M(n) is A007504(n).
%C A306457 The determinant of the matrix M(n) is A318173(n).
%C A306457 For n > 1, the subdiagonal sum of the matrix M(n) is A306192(n).
%H A306457 Stefano Spezia, <a href="/A306457/b306457.txt">Table of n, a(n) for n = 0..35</a>
%H A306457 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A306457 For n = 1 the matrix M(1) is
%e A306457     2
%e A306457 with permanent a(1) = 2.
%e A306457 For n = 2 the matrix M(2) is
%e A306457     2, 3
%e A306457     5, 2
%e A306457 with permanent a(2) = 19.
%e A306457 For n = 3 the matrix M(3) is
%e A306457     2, 3, 5
%e A306457     7, 2, 3
%e A306457    11, 7, 2
%e A306457 with permanent a(3) = 546.
%p A306457 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]);
%t A306457 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]
%o A306457 (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;}
%o A306457 a(n) = matpermanent(tm(n)); \\ _Michel Marcus_, Mar 16 2019
%Y A306457 Cf. A005843, A000040, A007504, A318173, A306192.
%K A306457 nonn
%O A306457 0,2
%A A306457 _Stefano Spezia_, Feb 17 2019
%E A306457 a(0) = 1 prepended by _Stefano Spezia_, Dec 06 2019