
This is too big to go into a comment...
A little bit of experimental observations:
The factorizations of these numbers are quite impressive. The primes which appear with a positive exponent appear with exponent $1$, except for two expections at most: $p$ and, in some cases, another prime, while typically there is only one. Moreover, these factors are, as far as I can caculate in small time, all small except for one huge factor. For example, factoring the sum for $p=163$ gives the following:
{{2,6},{3,4},{5,3},{7,1},{11,2},{13,1},{19,1},{23,1},{29,1},{37,1},{41,1},{43,1},{47,1},{53,1},{59,1},{61,1},{67,1},{71,1},{73,1},{79,1},{83,1},{89,1},{97,1},{101,1},{103,1},{107,1},{109,1},{113,1},{127,1},{131,1},{137,1},{139,1},{149,1},{151,1},{157,1},{163,2},{167,1},{173,1},{179,1},{181,1},{191,1},{193,1},{197,1},{199,1},{211,1},{223,1},{227,1},{229,1},{233,1},{239,1},{241,1},{251,1},{257,1},{263,1},{269,1},{271,1},{277,1},{281,1},{283,1},{293,1},{307,1},{311,1},{313,1},{317,1},{13220623261776675290879751941470274402307094729054895565509915203488199874013343384493,1}}
Here we see that $163$ is the only prime which appears more than one time, and all primes are $O(163)$ except the last one, which is pretty huge. (At that size, one really has to trust Wolfram!) This is typical (as far as I can compute in a small time, which is up to $163$ :) )
Moreover, as John Mangual observes in a comment above, this sums appear to be quite close to one third of the $p$th Catalan number. Mathematica quite immediately tells me that the sum is equal to $$\frac{\left(\begin{array}{c} 2 p \\\\ p\end{array}\right) {}\_3F_2\left(1,p,p+\frac{1}{2};p+1,p+1;4\right)}{p}\frac{2 i \pi }{3},$$ so if some asymtoptic information about ${}\_3F_2$ might actually prove this.


answered Jan 11 '10 at 2:46

