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).

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..18

Wikipedia, Skew-symmetric matrix

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

Cf. A000290, A004526, A083392.

Sequence in context: A024111 A231905 A203524 * A300426 A300688 A300611

Adjacent sequences: A331488 A331489 A331490 * A331492 A331493 A331494

Keyword

sign

Author

Stefano Spezia, Jan 18 2020

Status

approved