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.

1, 6, 180, 28296, 23762160, 103179627360, 2242514387116224, 244558402519846478976, 136585911664795732792710912, 392586698202941899973146848809472, 5721548125375080140228462836137111413760

Offset: 0

Author

Zhi-Wei Sun, Aug 14 2013

Keywords

• nonn

Comments

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.

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).

Examples

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..25

Crossrefs

Cf. A000172.

Programs

Mathematica

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}]