Bounds on the central binomial coefficient
by John from John D. Cook on (#3TTPG)
It's hard to find bounds on binomial coefficients that are both simple and accurate, but in 1990, E. T. H. Wang upper and lower bounds on the central coefficient that are both, valid for n at least 4.
Here are a few numerical results to give an idea of the accuracy of the bounds on a log scale. The first column is the argument n, The second is the log of the CBC (central binomial coefficient) minus the log of the lower bound. The third column is the log of the upper bound minus the log of the CBC.
|---+--------+--------| | n | lower | upper | |---+--------+--------| | 1 | 0.0000 | 0.3465 | | 2 | 0.0588 | 0.2876 | | 3 | 0.0793 | 0.2672 | | 4 | 0.0896 | 0.2569 | | 5 | 0.0958 | 0.2507 | | 6 | 0.0999 | 0.2466 | | 7 | 0.1029 | 0.2436 | | 8 | 0.1051 | 0.2414 | | 9 | 0.1069 | 0.2396 | |---+--------+--------|
And here's a plot of the same data taking n out further.
So the ratio of the upper bound to the CBC and the ratio of the CBC to the lower bound both quickly approach an asymptote, and the lower bound is better.