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A331491 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).

%I A331491 #9 Aug 20 2021 03:32:58
%S A331491 1,-1,8,-965,301864,-276973609,529706205072,-1976989515848629,
%T A331491 12817424808315680000,-136266429300554940901097,
%U A331491 2240244443768853657066332152,-54675928167021488863788002983045,1910142516402733768189592370043464696,-92787876901046051283841308281722409846473
%N A331491 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 A331491 Conjecture: for n > 0, det(M(2n)) = n^2 = A000290(n) with det(M(0)) = 1.
%H A331491 Vaclav Kotesovec, <a href="/A331491/b331491.txt">Table of n, a(n) for n = 0..18</a>
%H A331491 Wikipedia, <a href="https://en.wikipedia.org/wiki/Skew-symmetric_matrix">Skew-symmetric matrix</a>
%H A331491 Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A331491 For n = 2 the matrix M(4) is
%e A331491    0  1  1  2
%e A331491   -1  0  1  1
%e A331491   -1 -1  0  1
%e A331491   -2 -1 -1  0
%e A331491 with permanent a(2) = 8.
%t A331491 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 A331491 (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 A331491 a(n) = matpermanent(tm(2*n));
%Y A331491 Cf. A000290, A004526, A083392.
%K A331491 sign
%O A331491 0,3
%A A331491 _Stefano Spezia_, Jan 18 2020