= EGF coefficient
{c}
{title2=$[x^n]F(x)=a_n/n!$}
For an <exponential generating function> $F(x)=\sum_{n\geq0}a_nx^n/n!$, the ordinary coefficient operator gives $[x^n]F(x)=a_n/n!$. To recover the enumerated quantity, multiply by $n!$:
$$
a_n=n![x^n]F(x).
$$
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 $n=\sum_{k=0}^n\binom nk a_kb_{n-k}$. This follows by multiplying the two series and explains why <exponential generating functions> suit labeled <combinatorial structures>.
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