A228456 Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to A086618(i+j) for all i,j = 0,...,n.

1, 3, 41, 2841, 1010845, 1790959059, 15625174448153, 684155685766047665, 153135670584610711281133, 174306862433739623658457865851, 994128440039970508236558371536766105, 28386094077591547319667447446929656332136825

Offset: 0

Author

Zhi-Wei Sun, Aug 22 2013

Keywords

• nonn
• nice

Comments

Conjecture: (i) a(n) is always positive and odd, and not congruent to 7 modulo 8.

(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).

Examples

			a(0) = 1 since A086618(0) = 1.
		

References

• 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.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..25
• Zhi-Wei Sun, Conjectures and results on x^2 mod p^2 with 4p = x^2+d*y^2, arXiv:1103.4325 [math.NT], 2011-2014.
• Z.-W. Sun, On sums of Apéry polynomials and related congruences, J. Number Theory 132(2012), 2673-2699.
• Zhi-Wei Sun, Connections between p = x^2 + 3*y^2 and Franel numbers, J. Number Theory 133(2013), 2914-2928.
• Zhi-Wei Sun, On some determinants with Legendre symbol entries, preprint, arXiv:1308.2900 [math.NT], 2013-2018.

Crossrefs

Cf. A086618.

Programs

Mathematica

f[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k]/(k+1),{k,0,n}]
a[n_]:=Det[Table[f[i+j],{i,0,n},{j,0,n}]]
Table[a[n],{n,0,10}]

PARI

f(n)=sum(k=0,n, binomial(n,k)^2*binomial(2*k,k)/(k+1))
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