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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. 3

%I

%S 1,6,180,28296,23762160,103179627360,2242514387116224,

%T 244558402519846478976,136585911664795732792710912,

%U 392586698202941899973146848809472,5721548125375080140228462836137111413760

%N 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 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 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 Zhi-Wei Sun, <a href="/A225776/b225776.txt">Table of n, a(n) for n = 0..25</a>

%e a(0) = 1 since f(0+0) = 1.

%t 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 Cf. A000172.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Aug 14 2013

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Last modified October 28 10:53 EDT 2021. Contains 348327 sequences. (Running on oeis4.)