What's the Mathematica command for $\frac{\partial^a}{\partial n^a}\binom{n}{k}$?

Question

What's the Mathematica command for

$$f(a,n,k)=\frac{\partial^a}{\partial n^a}\binom{n}{k}$$

I tried

f[a_,n_,k_]:=D[Binomial[n,k], {n,a}] 

but didn't work. To make it work, I had to use

f[a_]:=D[Binomial[n,k], {n,a}]

then I picked a value for $a$ (f[2] for example) to get a function in terms of $n$ and $k$ then I selected values for $n$ and $k$.

Is there one-line command for this? Thank you.

Answer 1

The problem you face is, that you must evaluate the derivative relative to n before n is replaced by an argument. One way to achieve this is by using an additional variable: n1. E.g.:

f[a_, n_, k_] := D[Binomial[n1, k],{n1,a}] /. n1 -> n

with this:

f[1, 5, 3]
(* 47/6*)

Answer 2

Try

Derivative[a, 0][Function[{nn, kk}, Binomial[nn, kk]]][n, k]  

or

Derivative[a, 0][Binomial][n, k] (Thanks @CarlWoll )

$\frac{1}{2} \left( \begin{array}{cc} \{ & \begin{array}{cc} a! \left( \begin{array}{cc} \{ & \begin{array}{cc} 1 & a=2 \\ 2 n-1 & a=1 \\ \end{array} \\ \end{array} \right) & a\geq 1 \\ (n-1) n & \text{True} \\ \end{array} \\ \end{array} \right)$

Answer 3

Another way to do it: use Set rather than SetDelayed

g[a_, n_, k_] = D[Binomial[n, k], {n, a}];

The first derivative

g[1, 5, 3]

The second

g[2, 5, 3]