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 forĀ *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 argumentĀ *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.

Using Maple I found the following approximations:

U(n) = 4^n/(Pi*n)^(1/2),

L(n) = (1-1/(8*n)) * U(n).

These approximations get more accurate as n increases.

Based on some graphs, it look like U is an upper and L is a lower bound.

I suspect this may be proven using bounds for Stirling’s approximation [1].

[1] https://en.wikipedia.org/wiki/Stirling%27s_approximation#Speed_of_convergence_and_error_estimates