conformal field theory - Central charge in energy-momentum tensor OPE

I think that general point of view about central charge in books is considering OPE $T(z) T(w)$ for different field theories and finding that general expression for the most singular term is about to be $\sim \frac{c/2}{(z-w)^4}$. Then we do operator expansion $$T(z) = \sum \frac{L_n}{z^{n+2}}$$ and see that operators $L_n$ form Virasoro algebra. Or we can write transformation law for $T(z)$ and see that generally it's not a tensor, and here central charge comes into play again.

My question: I wonder if there is any way to start from Virasoro algebra generators and to prove that energy-momentum tensor in arbitrary theory must contain the term $\frac{c/2}{(z-w)^4}$ in its OPE with itself, where $c$ is a central charge of Virasoro algebra?

Another way to show this is to prove that operator expansion of $T(z)$ is always $\sum \frac{L_n}{z^{n+2}}$, where $L_n$ are Virasoro generators.