An Engel Transform

We can transform functions in the summands of an Engel expansion to sequence terms and or generating functions. A list gives E\left[{G(1+n)G(2+n)(2,2)_n}\right] \to A273935 = n!(n-1)!(2^n-1) \\ E\left[{G(1+n)G(2+n)}\right] \to A010790 = n!(n-1)! \\ E\left[{G(1+n)G(3+n)}\right] \to A129464 = -n(n+1)(n-1)!^2 \\ E\left[ 2^{-n^2-n+{24}} \pi ^{{2}+{4}} G(n+3)}{A^{3/2} G\left(n+{2}\right)}\right] = C(n) = {(n+1)!n!} \\ E\left[{\Gamma(n+1)}\right] \to 1,1,2,3,4,5,6,7,8,9, \cdots \,n \\ E\left[{\Gamma(n+1)^2}\right] \to 1,1,4,9,16,25,36,49,\cdots \,n^2 \\ E\left[{G(1+n)}\right] \to 1,1,1,2,6,24,120,720, \cdots \, n! \\ this is generated by the product of the reciprocals of the sequence terms ^n {a(k)} = E^{-1}(a(n))