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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
1, 6, 180, 28296, 23762160, 103179627360, 2242514387116224, 244558402519846478976, 136585911664795732792710912, 392586698202941899973146848809472, 5721548125375080140228462836137111413760 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

LINKS

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

EXAMPLE

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

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}]

CROSSREFS

Cf. A000172.

Sequence in context: A135395 A141121 A176730 * A051357 A251671 A064120

Adjacent sequences:  A225773 A225774 A225775 * A225777 A225778 A225779

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 14 2013

STATUS

approved

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)