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A322908 The determinant 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 A322908 #17 Nov 11 2020 09:00:31
%S A322908 1,-5,38,-386,4928,-75927,1371808,-28452356,666445568,-17402398505,
%T A322908 501297595904,-15792876550662,540190822408192,-19937252888438459,
%U A322908 789770307546718208,-33422580292067020808,1504926927960887066624,-71839548181524098808909,3624029163661165580910592
%N A322908 The determinant 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 A322908 The matrix M(n) differs from that of A318173 in using successive positive integers in place of successive prime numbers.
%C A322908 The trace of the matrix M(n) is A000027(n).
%C A322908 The sum of the first row of the matrix M(n) is A000217(n).
%C A322908 The sum of the first column of the matrix M(n) is A005448(n). [Corrected by _Stefano Spezia_, Dec 11 2019]
%C A322908 For n > 1, the sum of the superdiagonal of the matrix M(n) is A005843(n).
%H A322908 Vaclav Kotesovec, <a href="/A322908/b322908.txt">Table of n, a(n) for n = 1..300</a>
%H A322908 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A322908 a(n) ~ -(-1)^n * (3*exp(1) - exp(-1)) * n^n / 4. - _Vaclav Kotesovec_, Jan 05 2019
%e A322908 For n = 1 the matrix M(1) is
%e A322908    1
%e A322908 with determinant Det(M(1)) = 1.
%e A322908 For n = 2 the matrix M(2) is
%e A322908    1, 2
%e A322908    3, 1
%e A322908 with Det(M(2)) = -5.
%e A322908 For n = 3 the matrix M(3) is
%e A322908    1, 2, 3
%e A322908    4, 1, 2
%e A322908    5, 4, 1
%e A322908 with Det(M(3)) = 38.
%p A322908 a:= proc(n) uses LinearAlgebra;
%p A322908 Determinant(ToeplitzMatrix([seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))
%p A322908 end proc:
%p A322908 map(a, [$1..20]);
%t A322908 b[n_]:=n; a[n_]:=Det[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]; Array[a, 20]
%o A322908 (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 A322908 a(n) = matdet(tm(n)); \\ _Michel Marcus_, Nov 11 2020
%Y A322908 Cf. A000027, A000217, A005448, A005843, A318173.
%Y A322908 Cf. A322909 (permanent of matrix M(n)).
%K A322908 sign
%O A322908 1,2
%A A322908 _Stefano Spezia_, Dec 30 2018