cp's OEIS Frontend

A228379 Determinant of the n X n matrix with (i,j)-entry equal to (i^2+j^2)^n for all i,j = 0,...,n-1.

%I A228379 #19 Jul 14 2020 05:38:04
%S A228379 0,-1,-17280,1168415539200,980041972344422400000000,
%T A228379 -24517645963803990318633839493120000000000,
%U A228379 -37442952699741306217982755284947059704721771069440000000000000
%N A228379 Determinant of the n X n matrix with (i,j)-entry equal to (i^2+j^2)^n for all i,j = 0,...,n-1.
%C A228379 Conjecture: If n > 2, then (-1)^{n(n-1)/2}*a(n) > 0 and 2*Product_{k=1..n} (k!*(2k-1)!) divides a(n).
%C A228379 This conjecture implies that if p = 2*n-1 > 3 is a prime then we have a((p+1)/2) == 0 (mod p). So Sun's Conjecture 3.7 in the reference would follow from the above conjecture.
%C A228379 If b(n) denotes the n X n determinant with (i,j)-entry equal to (i+j)^n for all i,j = 0,...,n-1, then we conjecture that b(n)*(-1)^(n*(n-1)/2) / ((n-2)!*n*Product_{k=1..n} (k!)) is a positive integer for any integer n > 2.
%H A228379 Darij Grinberg, Zhi-Wei Sun, and Lilu Zhao, <a href="https://arxiv.org/abs/2007.06453">Proof of three conjectures on determinants related to quadratic residues</a>, arXiv:2007.06453 [math.NT], 2020.
%H A228379 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1308.2900">On some determinants with Legendre symbol entries</a>, preprint, arXiv:1308.2900 [math.NT], 2013-2019.
%t A228379 a[n_]:=Det[Table[(i^2+j^2)^n,{i,0,n-1},{j,0,n-1}]]
%t A228379 Table[a[n],{n,1,7}]
%o A228379 (PARI) a(n) = matdet(matrix(n, n, i, j, ((i-1)^2+(j-1)^2)^n)); \\ _Michel Marcus_, Jul 13 2020
%Y A228379 Cf. A227609.
%K A228379 sign
%O A228379 1,3
%A A228379 _Zhi-Wei Sun_, Aug 21 2013