Define

These binomial coefficients come up frequently in application. In particular, they came up in the previous post. I wanted to give an asymptotic approximation for *f*(*n*, *k*), and I thought it might be generally useful, so I pulled it out into its own post.

I used Mathematica to calculate an approximation. First, I used Stirling’s approximation.

s[n_] := Sqrt[2 Pi n] (n/E)^n

and then I used it to approximate *f*(*n*, *k*).

Simplify[ s[k n] / (s[(k-1) n] s[n]), Assumptions -> Element[k, Integers] && k > 0]

Applying `TeXForm`

to this gives

which is still a little hard to use.

By rearranging the terms we can make this approximation easier to understand.

These three terms are arranged in order of increasing importance. If *k* is fixed and *n* grows, the first term is constant, the second term decreases slowly, and the third term grows exponentially.

When *k* = 2, the approximation is particularly simple.

For larger the expressions are a little more complicated, but a couple examples will illustrate the pattern.

and

For an application of *f*(*n*,2), see How many ways you can parenthesize an expression?

For an application of *f*(*n*, 3) and *f*(*n*, 4), see the previous post.

John,

You forgot the exponent at the end of s[n_], should be (n/E)^n.

That’s all. Love your blogs! <(not factorial)

– Dan

Thanks! I was afraid the computations were wrong, but looks like I just left off the

`^n`

when I pasted the code.