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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).
2
1, -1, 8, -965, 301864, -276973609, 529706205072, -1976989515848629, 12817424808315680000, -136266429300554940901097, 2240244443768853657066332152, -54675928167021488863788002983045, 1910142516402733768189592370043464696, -92787876901046051283841308281722409846473
OFFSET
0,3
COMMENTS
Conjecture: for n > 0, det(M(2n)) = n^2 = A000290(n) with det(M(0)) = 1.
LINKS
Wikipedia, Toeplitz Matrix
EXAMPLE
For n = 2 the matrix M(4) is
0 1 1 2
-1 0 1 1
-1 -1 0 1
-2 -1 -1 0
with permanent a(2) = 8.
MATHEMATICA
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}]
PROG
(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; }
a(n) = matpermanent(tm(2*n));
CROSSREFS
KEYWORD
sign
AUTHOR
Stefano Spezia, Jan 18 2020
STATUS
approved