|
%I #20 Dec 08 2018 11:50:41
%S 1,3,41,2841,1010845,1790959059,15625174448153,684155685766047665,
%T 153135670584610711281133,174306862433739623658457865851,
%U 994128440039970508236558371536766105,28386094077591547319667447446929656332136825
%N Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to A086618(i+j) for all i,j = 0,...,n.
%C Conjecture: (i) a(n) is always positive and odd, and not congruent to 7 modulo 8.
%C (ii) For any odd prime p, if p == 1 (mod 3) and p = x^2 + 3*y^2 with x == 1 (mod 3), then a(p-1) == (-1)^{(p-1)/2}*(2*x-p/(2*x)) (mod p^2); if p == 2 (mod 3) then a(p-1) == (-1)^{(p+1)/2}*3p/binomial((p+1)/2,(p+1)/6) (mod p^2).
%D Zhi-Wei Sun, Conjectures and results on x^2 mod p^2 with 4p = x^2+d*y^2, in: Number Theory and Related Area (eds., Y. Ouyang, C. Xing, F. Xu and P. Zhang), Higher Education Press & International Press, Beijing and Boston, 2013, pp. 147-195.
%H Zhi-Wei Sun, <a href="/A228456/b228456.txt">Table of n, a(n) for n = 0..25</a>
%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/1103.4325">Conjectures and results on x^2 mod p^2 with 4p = x^2+d*y^2</a>, arXiv:1103.4325 [math.NT], 2011-2014.
%H Z.-W. Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2012.05.014">On sums of Apéry polynomials and related congruences</a>, J. Number Theory 132(2012), 2673-2699.
%H Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.jnt.2013.02.014">Connections between p = x^2 + 3*y^2 and Franel numbers</a>, J. Number Theory 133(2013), 2914-2928.
%H 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], 2013-2018.
%e a(0) = 1 since A086618(0) = 1.
%t f[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k]/(k+1),{k,0,n}]
%t a[n_]:=Det[Table[f[i+j],{i,0,n},{j,0,n}]]
%t Table[a[n],{n,0,10}]
%o (PARI) f(n)=sum(k=0,n, binomial(n,k)^2*binomial(2*k,k)/(k+1))
%o a(n)=my(v=vector(2*n+1,k,f(k-1))); matdet(matrix(n+1,n+1,i,j,v[i+j-1])) \\ _Charles R Greathouse IV_, Jul 30 2016
%Y Cf. A086618.
%K nonn,nice
%O 0,2
%A _Zhi-Wei Sun_, Aug 22 2013
|
