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A323254 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.

%I A323254 #20 Dec 11 2019 02:19:53
%S A323254 1,7,58,614,8032,125757,2298208,48075148,1133554432,29756555315,
%T A323254 860884417024,27218972906226,933850899349504,34556209025624041,
%U A323254 1371957513591119872,58174957356247084568,2624017129323317493760,125454378698728779884895,6337442836338834419089408
%N A323254 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.
%C A323254 The trace of the matrix M(n) is A000384(n). [Corrected by _Stefano Spezia_, Dec 08 2019]
%C A323254 The sum of the first row of the matrix M(n) is A034856(n).
%C A323254 The sum of the first column of the matrix M(n) is A000326(n).
%C A323254 For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
%C A323254 For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).
%H A323254 Vaclav Kotesovec, <a href="/A323254/b323254.txt">Table of n, a(n) for n = 1..300</a>
%H A323254 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz matrix</a>
%F A323254 a(n) ~ (5*exp(1) + exp(-1)) * n^n / 4. - _Vaclav Kotesovec_, Jan 10 2019
%e A323254 For n = 1 the matrix M(1) is
%e A323254    1
%e A323254 with determinant Det(M(1)) = 1.
%e A323254 For n = 2 the matrix M(2) is
%e A323254    3, 1
%e A323254    2, 3
%e A323254 with Det(M(2)) = 7.
%e A323254 For n = 3 the matrix M(3) is
%e A323254    5, 2, 1
%e A323254    4, 5, 2
%e A323254    3, 4, 5
%e A323254 with Det(M(3)) = 58.
%t A323254 b[i_]:=i; a[n_]:=Det[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2,n}]], Join[{b[2*n-1]},Array[b, n-1, {n-1,1}]]]]; Array[a,20]
%o A323254 (PARI) tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;}
%o A323254 a(n) = matdet(tm(n)); \\ _Stefano Spezia_, Dec 11 2019
%Y A323254 Cf. A000290, A000326, A000384, A001105, A001792.
%Y A323254 Cf. A034856, A204235, A318173, A322908, A322909.
%Y A323254 Cf. A323255 (permanent of matrix M(n)).
%K A323254 nonn
%O A323254 1,2
%A A323254 _Stefano Spezia_, Jan 09 2019