Recall the three sequences,

Equivalently,

where is the *generalized hypergeometric function*. Then it is known that,

Beautiful, aren’t they? Since the numbers increase at a near-geometric rate (for example, as *n* goes to infinity), then the convergence is very fast.

We also have the nice evaluations,

with the *Riemann zeta function* and the more general *Hurwitz zeta function* ,

respectively. (Note that for , the Hurwitz reduces into the Riemann.) The expression for *p* = 5 in the paper here used *Dirichlet L-functions*, but a poster from mathstackexchange gave it in terms of the Hurwitz zeta. The one for *p* = 7 is from Mathworld’s article on *central binomial coefficients*.

However, none are known for *p* > 7 (as well as *p* = 6). Based on odd *p*, it is easy to assume that the next has the form,

where,

and the *six* are *integers*. One can use Mathematica’s *LatticeReduce* function (which employs an integer relations algorithm) to find them, if any exists. Unfortunately, it didn’t find any exact relation, nor for analogous forms for prime *p* = 11 or 13. Either my old version of Mathematica is just not powerful enough, or odd *p* > 7 do not have analogous forms to the ones above.

*Can you find the next in the family?*