How do I craft this constant to a specific accuracy?

An earlier post was migrated to Physics.SE and we determined that if this converged, we would have a constant close to the mass of the proton in kilograms. This constant is constructed by summing the reciprocals of the products of all the numbers between consecutive squares.

Per this:

SumConvergence[10^-26/Pochhammer[m^2 + 1, 2 m], m]


(* True *)

we have convergence and thus a constant. However, the following:

Sum[10^-26/Pochhammer[m^2 + 1, 2 m], {m, \[Infinity]}]


outruns the precision.

My question: Is there a way to use this sum to create the constant to a specified accuracy?

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To be sure, the common opinion on physics.se of the meaning of this sum is that it is a sheer coincidence that it is close to the proton mass in kg. – Sjoerd C. de Vries Nov 13 '12 at 11:17
@SjoerdC.deVries, I just got the Popular Question badge --- 1000 views and only 2 votes. Yes, they don't like it. – Fred Kline Nov 13 '12 at 17:32

Clearly, you can factor out the tiny portion of your constant, so:

NSum[1/Pochhammer[m^2 + 1, 2 m], {m, 1, ∞}, WorkingPrecision -> 50] 1*^-26
1.6726218229590580987863882056891582636342622102204*10^-27


OTOH,

...products of all the numbers between consecutive squares.

Product[j, {j, (k - 1)^2, k^2}]
Pochhammer[(-1 + k)^2, 2 k]


This doesn't resemble what you have.

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