Inverse of the Pascal Matrix

In my opinion this is a bit simpler to prove, if we interpret the matrix $P_n$ as a linear transformation.

Consider the space $V_n$ of polynomials of degree $\le n$ (over, say $\Bbb{Q}$, but you are welcome to use reals or complex numbers, or any other field actually).

The mapping $T: f(x)\mapsto f(x+1)$ for all $f(x)\in V_n$ is obviously linear. My key point is to observe that $P_n$ is the matrix $M_{\mathcal B}(T)$ of $T$ with respect to the obvious basis $\mathcal{B}=\{1,x,x^2,\ldots,x^n\}$ of $V_n$. This is because for any $k, 0\le k\le n$ the binomial formula says that $$T(x^k)=\sum_{\ell=0}^k\binom k\ell x^\ell.$$ From this we can read the coordinates of $T(x^k)$ with respect to $\mathcal{B}$, and see that those coincide the $k$th column of $P_n$ (numbered from $0$ to $n$). That's exactly what the claim was.

It is obvious that the inverse of $T$ is given by the recipe $ T^{-1}:f(x)\mapsto f(x-1). $ A similar application of the binomial formula shows that the matrix $M_{\mathcal B}(T^{-1})$ then is exactly your prescribed inverse of $P_n$ containing binomial coefficients - this time with alternating signs.

But, by basic linear algebra $$ M_{\mathcal B}(T^{-1})=M_{\mathcal B}(T)^{-1}. $$

By the suggestion of i707107 one can use \begin{equation} \binom{j}{k}\binom{k}{i} = \binom{j}{i}\binom{j-i}{k-i} \end{equation} which is useful since it leads to the sum only going over one of the binomial coefficients. It leads to \begin{align} \sum^j_{k=i} (-1)^{j+k} \binom{j}{k} \binom{k}{i} &= \sum^j_{k=i} (-1)^{j+k} \binom{j}{i}\binom{j-i}{k-i} \\ &= (-1)^j \binom{j}{i} \sum^j_{k=i} (-1)^{k} \binom{j-i}{k-i} \\ &= (-1)^{j} \binom{j}{i} \sum^{j-i}_{k=0} (-1)^{k+i} \binom{j-i}{k} \\ &= (-1)^{j+i} \binom{j}{i} (1 - 1)^{j-i} = \delta_{ij} \end{align} The identity used is easily proven by $$ \binom{j}{k}\binom{k}{i} = \frac{j!}{k!(j-k)!} \frac{k!}{i!(k-i)!} = \frac{j!}{i!(j-i)!} \frac{(j-i)!}{(j-k)!(k-i)!} = \binom{j}{i}\binom{j-i}{k-i} $$ Another way of showing the identity involving $\delta_{ij}$ is to start with $x^j$ and using the binomial thoerem twice $$ x^j = (x+0)^j = ((x+1) - 1 )^j = ~...~ = \sum^j_{i=0} \sum^j_{k=i} (-1)^{j+k} \binom{j}{k} \binom{k}{i} x^i $$ comparing coefficients of the two sides of the equation, then also leads to the desired result. You can find the whole calcultion here, along whith other examples of when adding 0 really counts.

In the following it's somewhat more convenient to equivalently consider the transpose $P_n^T$ of the matrix $P_n$ and its inverse.

We obtain for $n=3$

\begin{align*} P_3^T&= \begin{pmatrix} 1 & & & \\ 1 & 1 & & \\ 1 & 2 & 1 & \\ 1 & 3 & 3 & 1 \\ \end{pmatrix}=\left(\binom{i}{j}\right)_{0\leq i,j\leq 3} \\ \left(P_3^T\right)^{-1}&= \begin{pmatrix} \begin{array}{rrrr} 1 & & & \\ -1 & 1 & & \\ 1 & -2 & 1 & \\ -1 & 3 & 3 & 1 \\ \end{array} \end{pmatrix}=\left((-1)^{i+j}\binom{i}{j}\right)_{0\leq i,j\leq 3} \end{align*}

The elements of the Pascal matrix can be considered as coefficients of binomial inverse pairs. These are sequences $(a_i)_{0\leq i \leq n},(b_i)_{0\leq i\leq n}$ which are related for $n\geq 0$ via \begin{align*} a_n=\sum_{i=0}^n\binom{n}{i}b_i\qquad\text{resp.}\qquad b_n=\sum_{i=0}^n(-1)^{i+n}\binom{n}{i}a_i\tag{1} \end{align*}

One relation implies the other in (1). This can be seen via exponential generating functions.

Let $A(x)$ and $B(x)$ be two exponential generating functions

$$A(x)=\sum_{n=0}^\infty a_{n}\frac{x^n}{n!} \qquad\qquad\text{ and }\qquad\qquad B(x)=\sum_{n=0}^\infty b_{n}\frac{x^n}{n!}$$

The expressions in (1) are the coefficients of

\begin{align*} A(x)=B(x)e^x\qquad\qquad\text{resp.}\qquad\qquad B(x)=A(x)e^{-x}\tag{2} \end{align*} The relationship in (2) is obvious and so the relationship in (1) follows.

The most important characterization of the pair of inverse relation in (1) is the orthogonal relation the pair implies. This follows by substituting one of the pair into the other. We obtain for $n\geq 0$

\begin{align*} a_n&=\sum_{j=0}^n\binom{n}{j}b_j\\ &=\sum_{j=0}^n\binom{n}{j}\sum_{i=0}^j(-1)^{i+j}\binom{j}{i}a_i\\ &=\sum_{i=0}^n a_i\sum_{j=i}^n(-1)^{i+j}\binom{n}{j}\binom{j}{i} \end{align*} Hence the orthogonal relation is \begin{align*} \delta_{ni}=\sum_{j=i}^n(-1)^{i+j}\binom{n}{j}\binom{j}{i} \end{align*} with $\delta_{ni}$ the Kronecker Delta.