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A322909 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.

%I A322909 #48 Jan 07 2020 16:50:47
%S A322909 1,1,7,100,2840,129428,8613997,791557152,95921167710,14818153059968,
%T A322909 2842735387366627,663020104070865664,184757202542187563476,
%U A322909 60623405966739216871680,23135486197103263598936745,10160292704659539620791062528,5087671168376607498331875818106
%N A322909 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 A322909 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 A322909 The trace of M(n) is A000027(n).
%C A322909 The sum of the first row of M(n) is A000217(n).
%C A322909 The sum of the first column of M(n) is A005448(n). [Corrected by _Stefano Spezia_, Dec 19 2019]
%C A322909 For n > 1, the sum of the superdiagonal of M(n) is A005843(n).
%C A322909 For n > 0, the sum of the (k-1)-th superdiagonal of M(n) is A003991(n,k). - _Stefano Spezia_, Dec 29 2019
%C A322909 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 A322909 Stefano Spezia, <a href="/A322909/b322909.txt">Table of n, a(n) for n = 0..35</a>
%H A322909 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A322909 For n = 1 the matrix M(1) is
%e A322909    1
%e A322909 with permanent a(1) = 1.
%e A322909 For n = 2 the matrix M(2) is
%e A322909    1, 2
%e A322909    3, 1
%e A322909 with permanent a(2) = 7.
%e A322909 For n = 3 the matrix M(3) is
%e A322909    1, 2, 3
%e A322909    4, 1, 2
%e A322909    5, 4, 1
%e A322909 with permanent a(3) = 100.
%p A322909 with(LinearAlgebra):
%p A322909 a:= n-> `if`(n=0, 1, Permanent(ToeplitzMatrix([
%p A322909          seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))):
%p A322909 seq(a(n), n = 0 .. 15);
%t A322909 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 A322909 (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 A322909 a(n) = matpermanent(tm(n)); \\ _Stefano Spezia_, Dec 19 2019
%Y A322909 Cf. A000027, A000217, A003991, A005448, A005843, A120070, A306457, A322908 (determinant of M(n)).
%K A322909 nonn
%O A322909 0,3
%A A322909 _Stefano Spezia_, Dec 30 2018
%E A322909 a(0) = 1 prepended by _Stefano Spezia_, Dec 19 2019