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.