|
%I #20 Sep 08 2022 08:45:11
%S 0,-9,240,-6300,181440,-5821200,207567360,-8172964800,352864512000,
%T -16593453676800,844757641728000,-46306798060723200,
%U 2720119606364160000,-170493211041753600000,11359219476176732160000,-801737767492652390400000,59762476409805241712640000,-4691769415367001788620800000
%N Determinant of the symmetric n X n matrix A defined by A[i,j] = abs(i^2 - j^2) for 1 <= i,j <= n.
%H Vincenzo Librandi, <a href="/A085799/b085799.txt">Table of n, a(n) for n = 1..140</a>
%F From _Vaclav Kotesovec_, Jan 08 2019: (Start)
%F a(n) ~ -(-1)^n * 2^(2*n - 3/2) * n^(n+2) / exp(n).
%F Recurrence: (14*n - 27)*a(n) = -8*(n-1)*(7*n + 4)*a(n-1) - 36*(2*n - 3)*a(n-2).
%F (End)
%e From _Klaus Brockhaus_, Apr 28 2010: (Start)
%e a(5) = determinant(A) = 181440 where A is the matrix
%e [ 0 3 8 15 24]
%e [ 3 0 5 12 21]
%e [ 8 5 0 7 16]
%e [15 12 7 0 9]
%e [24 21 16 9 0] (End)
%p (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
%t 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 *)
%o (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
%o (PARI) a(n) = matdet(matrix(n, n, i, j, abs(i^2-j^2))); \\ _Michel Marcus_, Aug 14 2017
%Y Cf. A085750.
%K sign
%O 1,2
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 24 2003
%E More terms from _José María Grau Ribas_, Apr 17 2010
%E Edited by _N. J. A. Sloane_, Apr 21 2010 at the suggestion of _R. J. Mathar_
%E More terms from _Michel Marcus_, Aug 14 2017
| 