2 added 542 characters in body

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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.

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This is too big to go into a comment...

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!)