 EGF coefficient

For an exponential generating function $F(x)={\sum }_{n\ge 0}{a}_n{x}^n/n!$, the ordinary coefficient operator gives $[x^n]F(x)=a_n/n!$. To recover the enumerated quantity, multiply by $n!$:For $F(x)=e^{2x}$, the ordinary coefficient is $2^n/n!$, but the represented counting sequence is $a_n=2^n$. Products have the binomial-convolution rule $n={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a_k{b}_{n-k}$. This follows by multiplying the two series and explains why exponential generating functions suit labeled combinatorial structures.