How to generate a recurrent sequence

Question

How to generate this type of sequence? $$ f(n, x) = x f'(n-1, x) \hspace{2 mm}, f(0, x) = e^x$$ How do I evaluate it for numerical values for $x = 1$ or any number?

Answer 1

Something like :

NestList[x D[#, x] &, Exp[x], 3]

(* {E^x, E^x x, x (E^x + E^x x), x (E^x + E^x x + x (2 E^x + E^x x))} *)

NestList[x D[#, x] &, Exp[x], 3] /. x-> 1

(* {E, E, 2 E, 5 E} *)

Answer 2

Leonid's method here can be easily adapted to your example:

experimentX[0, x_] := E^x;
experimentX[n_Integer, x_] := 
  Module[{xl}, 
   Set @@ Hold[experimentX[n, xl_], xl D[experimentX[n - 1, xl], xl]];
   experimentX[n, x]];

Answer 3

This pair of definitions will do what you want:

Clear[f];
f[n_, x_] /; IntegerQ[n] && n >= 1 := f[n, x] = x D[f[n - 1, x], x] // Simplify;
f[0, x_] = E^x;

It uses "memoisation" to save recomputing earlier results.