Question:
Let $\mathbf{A}$ be a square matrix of side $(p+1)$, and coefficients
$$A_{ij} = \binom{p}{j} \left(1-\dfrac{i}{p}\right)^{p-j} \left(\dfrac{i}{p}\right)^{j} \label{1}\tag{1}$$
Is there an analitic expression for $\mathbf{A}^{-1}$?
Motivation:
Let $B(u)$ be a bezier function of degree $p$ defined on $\left[0, \ 1\right]$.
$$B(u) = \sum_{j=0}^{p} B_{j}(u) \cdot Q_{j} \label{2}\tag{2}$$ $$B_{j}(u) = \binom{p}{j} \left(1-u\right)^{p-j} \cdot u^{j} \label{3}\tag{3}$$
$B(u)$ interpolates $f(u)$ at equally distributed knots $u_{i} = \dfrac{i}{p}$
$$B\left(\dfrac{i}{p}\right) = f\left(\dfrac{i}{p}\right) \ \ \ \ \ \ \forall \ i=0, \ \cdots, \ p \label{4}\tag{4}$$
I need to find the control points $\mathbf{Q}$.
One way to find $\mathbf{Q}$ is by solving a linear system \eqref{6}.
$$\sum_{j=0}^{p} B_{j}\left(u_i\right) \cdot Q_{j} = f\left(u_i\right) \ \ \ \ \ \ \forall \ i = 0, \ \cdots, \ p \label{5}\tag{5}$$ $$\underbrace{\begin{bmatrix} B_{0}(u_0) & B_{1}(u_0) & \cdots & B_{p}(u_0) \\ B_{0}(u_1) & B_{1}(u_1) & \cdots & B_{p}(u_1) \\ \vdots & \vdots & \ddots & \vdots \\ B_{0}(u_p) & B_{1}(u_p) & \cdots & B_{p}(u_p) \\ \end{bmatrix}}_{\mathbf{A}}\underbrace{\begin{bmatrix} Q_0 \\ Q_1 \\ \vdots \\ Q_{p} \end{bmatrix}}_{\mathbf{Q}} = \underbrace{\begin{bmatrix} f(u_0) \\ f(u_1) \\ \vdots \\ f(u_p) \end{bmatrix}}_{\mathbf{F}} \label{6}\tag{6}$$
Problem: Solving the linear system using a computer is expensive.
Then I search for a mathematical expression for $\mathbf{A}$'s inverse.
$$\sum_{j=0}^{n} \underbrace{\binom{p}{j} \left(1-\dfrac{i}{p}\right)^{p-j} \left(\dfrac{i}{p}\right)^{j}}_{A_{ij}} \cdot Q_{j} = f\left(u_i\right)\ \ \ \ \ \forall i=0, \cdots, p \label{7}\tag{7}$$
The matrix you’re dealing with is called a Bernstein-Vandermonde matrix. For an analytic expression for its inverse, look at section 2.3 of this paper, or here.
But the inversion formulas are very complex, and I’m not sure they’re worth the effort. If you’re really concerned about performance, I’d recommend just hard-coding the matrix inverses. For each degree that interests you, compute the matrix inverse once, in exact arithmetic (by using Mathematica, for example), and store it in your code.
I don’t know how you’re going to use this interpolation function. But if you’re trying to approximate $f$ using a Bézier curve, then using equally spaced points $i/p$ is a really bad idea. Use Chebyshev nodes, instead. You’ll get a different family of Bernstein-Vandermonde matrices, but you can still pre-compute their inverses.