polynomials - Why does Expand not work within a function?

I'm writing this fairly simple function:

hermite[0, x] = 1;
hermite[1, x] = 2 x;
hermite[n_, x_] := (
  hermite[n, x] = 2 x*hermite[n - 1, x] - 2 (n - 1) hermite[n - 2, x];
  Expand[hermite[n, x]]
  )

But the Expand command is ignored.

Yet, when I do

Expand[hermite[10, x]]

The result is expanded like I wish.

Why is it not working when I put the same command inside a function? I'd like to do it in the function; it would be cleaner.