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A323255 The permanent 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 A323255 #20 Dec 11 2019 05:04:45
%S A323255 1,1,11,248,9968,638772,60061657,7798036000,1336715859150,
%T A323255 292406145227392,79483340339739367,26280500564448081664,
%U A323255 10386012861097225139356,4834639222955142417477888,2618110215141486526589786501,1631888040186649673361825042432,1159983453675106278249250918734938
%N A323255 The permanent 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 A323255 The trace of the matrix M(n) is A000384(n).
%C A323255 The sum of the first row of the matrix M(n) is A034856(n).
%C A323255 The sum of the first column of the matrix M(n) is A000326(n).
%C A323255 For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
%C A323255 For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).
%H A323255 Stefano Spezia, <a href="/A323255/b323255.txt">Table of n, a(n) for n = 0..35</a>
%H A323255 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz matrix</a>
%e A323255 For n = 1 the matrix M(1) is
%e A323255    1
%e A323255 with permanent a(1) = 1.
%e A323255 For n = 2 the matrix M(2) is
%e A323255    3, 1
%e A323255    2, 3
%e A323255 with permanent a(2) = 11.
%e A323255 For n = 3 the matrix M(3) is
%e A323255    5, 2, 1
%e A323255    4, 5, 2
%e A323255    3, 4, 5
%e A323255 with permanent a(3) = 248.
%t A323255 b[i_]:=i; a[n_]:=If[n==0, 1, Permanent[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, 16, 0]
%o A323255 (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 A323255 a(n) = matpermanent(tm(n)); \\ _Stefano Spezia_, Dec 11 2019
%Y A323255 Cf. A000290, A000326, A000384, A001105, A001792.
%Y A323255 Cf. A034856, A204235, A318173, A322908, A322909.
%Y A323255 Cf. A323254 (determinant of matrix M(n)).
%K A323255 nonn
%O A323255 0,3
%A A323255 _Stefano Spezia_, Jan 09 2019
%E A323255 a(0) = 1 prepended by _Stefano Spezia_, Dec 08 2019