# Difference Between Generating Functions And Exponential Generating Functions [duplicate]

What is the difference between generating functions and exponential generating functions?

I mean, we can solve a combinatorics sum by using just simple generating functions.

Or simply, for which parts in combinatorics do we use simple generating functions and in which parts we use exponential generating functions?

## marked as duplicate by Hans Lundmark, Namaste, darij grinberg, Math1000, ShaileshJan 16 at 0:01

For ordinary generating functions, that convolution is $$(a*b)_n = \sum_{j=0}^n a_j b_{n-j}$$ corresponding to the usual translation-invariant counting measure on the nonnegative integers. There are a great many situations in which this appears.
For exponential generating functions, that convolution is $$(a*b)_n = \sum_{j=0}^n \binom{n}{j} a_j b_{n-j}$$ This form, reminiscent of the binomial theorem, is something that occasionally comes up in combinatorics - and when it does, generating functions aren't going to work well unless we make that adjustment.