# Sum of ${m \choose j}$ multiplied by $2^{2^j}$

I'm trying to find an expression for

$$\sum_{j=1}^m {m \choose j} 2^{2^j}$$

I have tried doing a change of variables on $2^j$, and I've also tried to apply some identities that are similar to the Binomial identity (which I think I could use if the exponent was, say $(2^2)^j$ instead).

I suspect that maybe I could find the answer by subtracting two binomial sums, because this sum essentially "skips" some terms, but I haven't succeeded.