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.

Thank you in advance!


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