A322909 - OEIS

%I #48 Jan 07 2020 16:50:47

%S 1,1,7,100,2840,129428,8613997,791557152,95921167710,14818153059968,

%T 2842735387366627,663020104070865664,184757202542187563476,

%U 60623405966739216871680,23135486197103263598936745,10160292704659539620791062528,5087671168376607498331875818106

%N 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.

%C 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_]

%C The trace of M(n) is A000027(n).

%C The sum of the first row of M(n) is A000217(n).

%C The sum of the first column of M(n) is A005448(n). [Corrected by _Stefano Spezia_, Dec 19 2019]

%C For n > 1, the sum of the superdiagonal of M(n) is A005843(n).

%C For n > 0, the sum of the (k-1)-th superdiagonal of M(n) is A003991(n,k). - _Stefano Spezia_, Dec 29 2019

%C 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

%H Stefano Spezia, <a href="/A322909/b322909.txt">Table of n, a(n) for n = 0..35</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>

%e For n = 1 the matrix M(1) is

%e    1

%e with permanent a(1) = 1.

%e For n = 2 the matrix M(2) is

%e    1, 2

%e    3, 1

%e with permanent a(2) = 7.

%e For n = 3 the matrix M(3) is

%e    1, 2, 3

%e    4, 1, 2

%e    5, 4, 1

%e with permanent a(3) = 100.

%p with(LinearAlgebra):

%p a:= n-> `if`(n=0, 1, Permanent(ToeplitzMatrix([

%p          seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))):

%p seq(a(n), n = 0 .. 15);

%t 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]

%o (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; }

%o a(n) = matpermanent(tm(n)); \\ _Stefano Spezia_, Dec 19 2019

%Y Cf. A000027, A000217, A003991, A005448, A005843, A120070, A306457, A322908 (determinant of M(n)).

%K nonn

%O 0,3

%A _Stefano Spezia_, Dec 30 2018

%E a(0) = 1 prepended by _Stefano Spezia_, Dec 19 2019