grassmannians – A specific integration with Grassmann variables Answer

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grassmannians – A specific integration with Grassmann variables

I have recently read (for example, here) that this relation below is true

$$ \int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}), $$
where $Pf(\mathbf{A})$ is the Pfaffian of an even dimensional skew-(or anti-)symmetric matrix $\mathbf{A}$ other $\{ z_i \}$ are Grassman variables.

Could anyone point out if there is a similar formula for the form below:

$$ \int dz \: \big( \frac{1}{2} \sum_{ij} z_i A_{ij} z_j \big)^k, $$
where $k > 0$ is an integer?

My guess is yes since

$$ e^{\mathbf{A}} = \sum_{j=0}^\infty \frac{\mathbf{A}^j}{j!}, $$
but I do not have enough experience with these types of integrals.

Thanks in advance!

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