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A307783 The permanent of an n X n symmetric Toeplitz matrix M(n) whose first row consists of n, n-1, ..., 1.

%I A307783 #15 Jan 06 2023 09:28:07
%S A307783 1,5,62,1472,57228,3300052,264163120,28004426240,3796084024832,
%T A307783 640290996560896,131495036625989504,32300689159458652160,
%U A307783 9350873610168606862080,3150550820854335942423808,1222211647879605626853439488,540858935979668390014623285248,270804098518125729769134021574656
%N A307783 The permanent of an n X n symmetric Toeplitz matrix M(n) whose first row consists of n, n-1, ..., 1.
%C A307783 The matrix M(n) differs from that of A204235 in using for the first row the positive integers 1, 2,..., n in decreasing order in place of in increasing order (see examples).
%C A307783 The trace of the matrix M(n) is A000290(n).
%C A307783 The determinant of the matrix M(n) is A001792(n-1).
%C A307783 The sum of the k-th row of the matrix M(n) is A008867(n,k).
%C A307783 For n > k, the sum of the k-diagonal of the matrix M(n) is A055461(n,k).
%H A307783 Vaclav Kotesovec, <a href="/A307783/b307783.txt">Table of n, a(n) for n = 1..35</a>
%H A307783 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A307783 For n = 1 the matrix M(1) is
%e A307783   1
%e A307783 with permanent a(1) = 1.
%e A307783 For n = 2 the matrix M(2) is
%e A307783   2, 1
%e A307783   1, 2
%e A307783 with permanent a(2) = 5.
%e A307783 For n = 3 the matrix M(3) is
%e A307783   3, 2, 1
%e A307783   2, 3, 2
%e A307783   1, 2, 3
%e A307783 with permanent a(3) = 62.
%p A307783 f:= proc(n) uses LinearAlgebra; Permanent(ToeplitzMatrix([i, i=n..1, -1)])) end proc: map(f, [$1..17]);
%t A307783 b[i_]:=i; a[n_]:=Permanent[ToeplitzMatrix[Reverse[Array[b, n]], Reverse[Array[b, n ]]]]; Array[a, 17]
%o A307783 (PARI) {a(n) = matpermanent(matrix(n, n, i, j, n + 1 - max(i - j + 1, j - i + 1)))}
%o A307783 for(n=1, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Apr 29 2019
%Y A307783 Cf. A000290, A001792, A008867, A055461, A204235.
%K A307783 nonn
%O A307783 1,2
%A A307783 _Stefano Spezia_, Apr 28 2019