Polynomial Laplace transform
Haskell, 25 bytes

zipWith(*)$scanl(*)1[1..]

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Pretty straightforward: Generates the list of factorials [1,1,2,6,...] with a scanl, then does zipWith(*) to multiply each element of the input by the corresponding value.
32 bytes

foldr(\(i,x)r-&gt;x:map((i+1)*)r)[]

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A pretty folding-based solution. Takes inputs as (exponent, coefficient) pairs.