= Solution
In any two-dimensional conformal field theory,
$$
T(z)T(w)\sim\frac{c/2}{(z-w)^4}
+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{z-w}.
$$
Double contractions in the bosonic <beta-gamma system> give
$$
\boxed{c_{\beta\gamma}=2(6h^2-6h+1)}.
$$
The polynomial is unchanged by $h\mapsto1-h$, as required when the roles of the two fields are interchanged. If the fields anticommute, a closed fermionic contraction contributes an additional minus sign, giving the fermionic bc-system result
$$
\boxed{c_{bc}=-2(6h^2-6h+1)}.
$$
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