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

1,2

Links

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

Sela Fried, On sequence A085799, 2026.

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)

For n>=2, a(n) = (-1)^(n-1) * (n+1)/2 * (2*n-1)! / (n-2)!. Furthermore, the conjecture by Lajos is true (see Fried link). - Sela Fried, Feb 04 2026

Example

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)

Maple

# Conjectured to give the same sequence, apart from signs
with(combstruct):
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 *)

Prog

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

Crossrefs

Cf. A085750.

Sequence in context: A279492 A297747 A211052 * A278858 A274789 A263158

Adjacent sequences: A085796 A085797 A085798 * A085800 A085801 A085802

Keyword

sign

Author

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

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

Status

approved