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A225776 Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to f(i+j) for all i,j = 0,...,n, where f(k) = A000172(k) is the k-th Franel number.

%I A225776 #71 Feb 11 2015 23:12:23
%S A225776 1,6,180,28296,23762160,103179627360,2242514387116224,
%T A225776 244558402519846478976,136585911664795732792710912,
%U A225776 392586698202941899973146848809472,5721548125375080140228462836137111413760
%N A225776 Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to f(i+j) for all i,j = 0,...,n, where f(k) = A000172(k) is the k-th Franel number.
%C A225776 Conjecture: a(n)/6^n is always a positive odd integer. Moreover, for any integers r > 1 and n >= 0, the number a(r,n)/2^n is a positive odd integer, where a(r,n) denotes the Hankel determinant |f(r,i+j)|_{i,j=0,...,n} with f(r,k) = sum_{j=0}^k C(k,j)^r.
%C A225776 On Aug 20 2013, _Zhi-Wei Sun_ made the following conjecture: If p is a prime congruent to 1 mod 4 but p is not congruent to 1 mod 24, then p divides a((p-1)/2).
%H A225776 Zhi-Wei Sun, <a href="/A225776/b225776.txt">Table of n, a(n) for n = 0..25</a>
%e A225776 a(0) = 1 since f(0+0) = 1.
%t A225776 f[n_]:=Sum[Binomial[n,k]^3,{k,0,n}]; a[n_]:=Det[Table[f[i+j],{i,0,n},{j,0,n}]]; Table[a[n],{n,0,10}]
%Y A225776 Cf. A000172.
%K A225776 nonn
%O A225776 0,2
%A A225776 _Zhi-Wei Sun_, Aug 14 2013