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a(n) is the permanent of a 2n X 2n antisymmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).
2

%I #9 Aug 20 2021 03:32:58

%S 1,-1,8,-965,301864,-276973609,529706205072,-1976989515848629,

%T 12817424808315680000,-136266429300554940901097,

%U 2240244443768853657066332152,-54675928167021488863788002983045,1910142516402733768189592370043464696,-92787876901046051283841308281722409846473

%N a(n) is the permanent of a 2n X 2n antisymmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

%C Conjecture: for n > 0, det(M(2n)) = n^2 = A000290(n) with det(M(0)) = 1.

%H Vaclav Kotesovec, <a href="/A331491/b331491.txt">Table of n, a(n) for n = 0..18</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Skew-symmetric_matrix">Skew-symmetric matrix</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>

%e For n = 2 the matrix M(4) is

%e 0 1 1 2

%e -1 0 1 1

%e -1 -1 0 1

%e -2 -1 -1 0

%e with permanent a(2) = 8.

%t nmax:=13; k[i_]:=Floor[i/2]; a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[-Array[k, n], Array[k, n]]]]; Table[a[2n],{n,0,nmax}]

%o (PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, floor(j/2), if (j==1, -floor(i/2))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }

%o a(n) = matpermanent(tm(2*n));

%Y Cf. A000290, A004526, A083392.

%K sign

%O 0,3

%A _Stefano Spezia_, Jan 18 2020