A085799 Determinant of the symmetric n X n matrix A defined by A[i,j] = abs(i^2 - j^2) for 1 <= i,j <= n.

0, -9, 240, -6300, 181440, -5821200, 207567360, -8172964800, 352864512000, -16593453676800, 844757641728000, -46306798060723200, 2720119606364160000, -170493211041753600000, 11359219476176732160000, -801737767492652390400000, 59762476409805241712640000, -4691769415367001788620800000

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 24 2003

Keywords

• sign

Examples

			From _Klaus Brockhaus_, Apr 28 2010: (Start)
a(5) = determinant(A) = 181440 where A is the matrix
  [ 0  3  8 15 24]
  [ 3  0  5 12 21]
  [ 8  5  0  7 16]
  [15 12  7  0  9]
  [24 21 16  9  0] (End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..140

Crossrefs

Cf. A085750.

Programs

Magma

[ Determinant( SymmetricMatrix( &cat[ [ Abs(i^2-j^2): j in [1..i] ]: i in [1..n] ] ) ): n in [1..15] ]; // Klaus Brockhaus, Apr 28 2010

Maple

(Conjectured to give the same sequence, apart from signs): a:=n->sum((count(Permutation(n*2-1),size=n+1)),j=0..n)/2: seq(a(n), n=1..16); # Zerinvary Lajos, May 03 2007

Mathematica

A[i_, j_] := Abs[i^2 - j^2]; a[n_] := Det[Table[A[i, j], {i, n}, {j, n}]]; Table[a[n], {n, 44}] (* José María Grau Ribas, Apr 17 2010 *)

PARI

a(n) = matdet(matrix(n, n, i, j, abs(i^2-j^2))); \\ Michel Marcus, Aug 14 2017

Formula

From Vaclav Kotesovec, Jan 08 2019: (Start)

a(n) ~ -(-1)^n * 2^(2*n - 3/2) * n^(n+2) / exp(n).

Recurrence: (14*n - 27)*a(n) = -8*(n-1)*(7*n + 4)*a(n-1) - 36*(2*n - 3)*a(n-2).

(End)

Extensions

More terms from José María Grau Ribas, Apr 17 2010

Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar

More terms from Michel Marcus, Aug 14 2017