I found a rough correspondence between the formula under the Consider: and Pascal's triangle...

Also, one can solve for the middle number approximately:

for r = 11

the right answer is 252 so pretty close.

One practical reason for this approach is for example finding the middle number of row r of pascal's triangle, ordinarily you would use the combinatoric formula r!/((r-1!)*(r!)) but eventually the factorial involves numbers too large for Maple to calculate, but using my integral formula for the middle number only involves calculating 2^r which it can do more easily.

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