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A228289 Determinant of the p_n X p_n matrix with (i,j)-entry equal to D(i+j) for all i,j = 0,...,p_n-1, where D(k) = A002895(k) is the k-th Domb number and p_n is the n-th prime.

%I A228289 #12 Oct 06 2021 17:56:26
%S A228289 12,2448,428587718400,4994319435309277891448832,
%T A228289 191901511752240055024005979549622856313555581586068578283027431424,
%U A228289 637213222716753775758429677219909335140503764595701930312765250413280716374852064945052319744
%N A228289 Determinant of the p_n X p_n matrix with (i,j)-entry equal to D(i+j) for all i,j = 0,...,p_n-1, where D(k) = A002895(k) is the k-th Domb number and p_n is the n-th prime.
%C A228289 Conjecture: If p_n == 1 (mod 3) and p_n = x^2 + 3*y^2 with x and y integers, then we have a(n) == (-1)^{(p_n-1)/2}*(4*x^2-2*p_n) (mod p_n^2). In the case p_n == 2 (mod 3), we have a(n) == 0 (mod p_n^2).
%C A228289 Zhi-Wei Sun also made the following similar conjecture:
%C A228289 If p is an odd prime and b(p) is the p X p determinant with (i,j)-entry equal to A053175(i+j) for all i,j = 0,...,p-1, then we have the congruence b(p) == (-1)^{(p-1)/2} (mod p^2).
%D A228289 Zhi-Wei Sun, Conjectures and results on x^2 mod p^2 with 4*p = 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.
%t A228289 d[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k]Binomial[2(n-k),n-k],{k,0,n}]
%t A228289 a[n_]:=Det[Table[d[i+j],{i,0,Prime[n]-1},{j,0,Prime[n]-1}]]
%t A228289 Table[a[n],{n,1,8}]
%Y A228289 Cf. A002895, A053175, A225776, A228143.
%K A228289 nonn
%O A228289 1,1
%A A228289 _Zhi-Wei Sun_, Aug 19 2013