One should be able to obtain the formula from the appendix of:

http://arxiv.org/abs/math-ph/0204029

They have a formula for all vectors, to the vacuum expectation just the summand with $2p=n$ contributes. They using Arakis self dual CAR algebra, and if you consider $a(f)$ for $f=\Gamma f$ it should equal your $c(f)+c(f)^\ast$.

One should be able to obtain the formula from the appendix of:

• Hellmut Baumgärtel, Matthias Jurke, Fernando Lledó, Twisted duality of the CAR-Algebra, J.Math.Phys. 43 (2002) 4158-4179, https://doi.org/10.1063/1.1483376, http://arxiv.org/abs/math-ph/0204029

They have a formula for all vectors, to the vacuum expectation just the summand with $2p=n$ contributes. They using Arakis self dual CAR algebra, and if you consider $a(f)$ for $f=\Gamma f$ it should equal your $c(f)+c(f)^\ast$.