How would I find the series generated by $e^x$ + $4x^2$? (exponential generating function)

Question

Can I simply say: $e^x$ generates 1 + x/1! + $x^2$/2! + .... and add on the $4x^2$ term? so the sequence generated instead of being 1,1,1,1,1,... is 1,1,4,1,1,1...?

Answer 1

Recall, an exponential generating function has the representation \begin{align*} \sum_{n=0}^\infty a_n\frac{x^n}{n!}\tag{1} \end{align*}

The expression $4x^2$ can be written as $$8\frac{x^2}{2!}$$ to fit the representation in (1). The sequence of coefficients of the exponential generating function $e^x+4x^2$ is therefore \begin{align*} 1,1,\color{blue}{9},1,1,1,\ldots \end{align*}