Mathematica: reformulating polynomial coefficients

Question

I am looking for a way to put the polynomial:

x + 5x^2/2 + 3x^3 + 8x^4/3 + 43x^5/24 + 43x^6/48

into a more "Taylor"-ish form:

x/1! + 5x^2/2! + 18x^3/3! + 64x^4/4! + 215x^5/5! + 645x^6/6!

My main goal is to be able to read off the coefficients from the new form. i.e. I am interested in the numbers:

1,5,18,64,215,645 etc..

Answer 1

This?

poly = x + 5 x^2/2 + 3 x^3 + 8 x^4/3 + 43 x^5/24 + 43 x^6/48;
Table[poly[[i]]/x^i*i!, {i, Length[poly]}]

which gives

{1, 5, 18, 64, 215, 645}