Find $a_n$ in terms of $b_n$ given $b_n = \sum_{k=0}^{n} {n \choose k} a_k$

Question

Given sequences $a_n$ and $b_n$ satifying
$$b_n = \sum_{k=0}^{n} {n \choose k} a_k$$

I am required to find $a_n$ in terms of $b_n$

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My attempt:

The generating fuction for $b_n$ will be
\begin{align}
B(x) &= \sum_{n=0}^{\infty} \left( \sum_{k=0}^{n} {n \choose k} a_k \right)\\
&= \sum_{n=0}^{\infty} \left( \sum_{k=0}^{n} {n \choose n-k} a_k \right)
\end{align}
This looks like the product of two generating functions $A(x)$ ( for $a_n$ ) and $C(x)$. Hence the given sequence $b_n$ is an convolution of two $a_n$ and some $c_n$.

If now I can find $c_n$, and a closed form for $C(x)$ (which I believe exists), the sequence $a_n$ can be found since $$A(x) = \frac{B(x)}{C(x)}$$

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My question:

I am unable to find the sequence $c_n$. I tried using $c_k = {n \choose k}$ but I am quite sure that it is incorrect.

Answer 1

Here we are looking for so-called binomial inverse pairs. To show the relationship we multiply exponential generating functions (egfs). Let $A(x)=\sum_{n\ge0}a_{n}\frac{x^n}{n!}$ and $B(x)=\sum_{n\ge0}b_{n}\frac{x^n}{n!}$ egfs with $B(x)=A(x)e^x$. Comparing coefficients gives the following

Binomial inverse pair \begin{align*} B(x)&=A(x)e^x&A(x)&=B(x)e^{-x}\\ b_n&=\sum_{k=0}^{n}\binom{n}{k}a_k&a_n&=\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}b_k \end{align*}