Answer by Slaviks for Correlated three-particle Green Function

I was able to confirm both of your formulas, assuming that the operators are fermionic and normal ordered.

Here is the general algorithm for generating a cumulant average of $n$ variables (taken from page page 34 of G.W.Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, attributed to van Kampen):

$$G_c(X_1,X_2,\ldots, X_n)=\sum_{p=0}^{n-1}(-1)^p C_p(X_1,X_2,\ldots,X_n)$$where $$C_p(X_1,X_2,\ldots,X_n)=G(X_1) G(X_2,X_3,\ldots,X_n)+ \ldots$$ is the sum over all distinct partitions of $n$ variables into $p$ subsets, with the averaging function $G$ applied to each subset. If the operators are fermionic, then each term in $C_p$ has to be multiplied by the sign of the required permutation.

Here are references to the original papers:

1. E. Meeron, J. Chem. Phys. 27, 1238 (1957); DOI:10.1063/1.1743985
2. N.G. van Kampen, Physica 74, 215 (1973); 74, 239 (1973.)

Now the computer algebra: using this answer, it is easy to write a Mathematica code computing $C_p$ and thus $G_c$ for the general case.

Assuming that all your variables up to 3 are creation operators, and from 4 to 6 are are annihilation operators (or vice versa), and that the particle number is well-defined, we have to keep averages only of an equal number of creation and annihilation operators. In this way I can confirm the validity of each term in both formulas quoted.

As "proof of work" I can offer you the connected function of 8 variables (hope it does not break MathJax in your browser):$$G_c(1,2,3,4,5,6,7,8)=-6 G(1,8) G(2,7) G(3,6) G(4,5)+6 G(1,7) G(2,8) G(3,6) G(4,5)+6 G(1,8) G(2,6) G(3,7) G(4,5)-6 G(1,6) G(2,8) G(3,7) G(4,5)-6 G(1,7) G(2,6) G(3,8) G(4,5)+6 G(1,6) G(2,7) G(3,8) G(4,5)+2 G(3,8) G(1,2,6,7) G(4,5)-2 G(3,7) G(1,2,6,8) G(4,5)+2 G(3,6) G(1,2,7,8) G(4,5)-2 G(2,8) G(1,3,6,7) G(4,5)+2 G(2,7) G(1,3,6,8) G(4,5)-2 G(2,6) G(1,3,7,8) G(4,5)+2 G(1,8) G(2,3,6,7) G(4,5)-2 G(1,7) G(2,3,6,8) G(4,5)+2 G(1,6) G(2,3,7,8) G(4,5)-G(1,2,3,6,7,8) G(4,5)+6 G(1,8) G(2,7) G(3,5) G(4,6)-6 G(1,7) G(2,8) G(3,5) G(4,6)-6 G(1,8) G(2,5) G(3,7) G(4,6)+6 G(1,5) G(2,8) G(3,7) G(4,6)+6 G(1,7) G(2,5) G(3,8) G(4,6)-6 G(1,5) G(2,7) G(3,8) G(4,6)-6 G(1,8) G(2,6) G(3,5) G(4,7)+6 G(1,6) G(2,8) G(3,5) G(4,7)+6 G(1,8) G(2,5) G(3,6) G(4,7)-6 G(1,5) G(2,8) G(3,6) G(4,7)-6 G(1,6) G(2,5) G(3,8) G(4,7)+6 G(1,5) G(2,6) G(3,8) G(4,7)+6 G(1,7) G(2,6) G(3,5) G(4,8)-6 G(1,6) G(2,7) G(3,5) G(4,8)-6 G(1,7) G(2,5) G(3,6) G(4,8)+6 G(1,5) G(2,7) G(3,6) G(4,8)+6 G(1,6) G(2,5) G(3,7) G(4,8)-6 G(1,5) G(2,6) G(3,7) G(4,8)+2 G(3,8) G(4,7) G(1,2,5,6)-2 G(3,7) G(4,8) G(1,2,5,6)-2 G(3,8) G(4,6) G(1,2,5,7)+2 G(3,6) G(4,8) G(1,2,5,7)+2 G(3,7) G(4,6) G(1,2,5,8)-2 G(3,6) G(4,7) G(1,2,5,8)-2 G(3,5) G(4,8) G(1,2,6,7)+2 G(3,5) G(4,7) G(1,2,6,8)-2 G(3,5) G(4,6) G(1,2,7,8)-2 G(2,8) G(4,7) G(1,3,5,6)+2 G(2,7) G(4,8) G(1,3,5,6)+2 G(2,8) G(4,6) G(1,3,5,7)-2 G(2,6) G(4,8) G(1,3,5,7)-2 G(2,7) G(4,6) G(1,3,5,8)+2 G(2,6) G(4,7) G(1,3,5,8)+2 G(2,5) G(4,8) G(1,3,6,7)-2 G(2,5) G(4,7) G(1,3,6,8)+2 G(2,5) G(4,6) G(1,3,7,8)+2 G(2,8) G(3,7) G(1,4,5,6)-2 G(2,7) G(3,8) G(1,4,5,6)-2 G(2,8) G(3,6) G(1,4,5,7)+2 G(2,6) G(3,8) G(1,4,5,7)+2 G(2,7) G(3,6) G(1,4,5,8)-2 G(2,6) G(3,7) G(1,4,5,8)+2 G(2,8) G(3,5) G(1,4,6,7)-2 G(2,5) G(3,8) G(1,4,6,7)-2 G(2,7) G(3,5) G(1,4,6,8)+2 G(2,5) G(3,7) G(1,4,6,8)+2 G(2,6) G(3,5) G(1,4,7,8)-2 G(2,5) G(3,6) G(1,4,7,8)+2 G(1,8) G(4,7) G(2,3,5,6)-2 G(1,7) G(4,8) G(2,3,5,6)-G(1,4,7,8) G(2,3,5,6)-2 G(1,8) G(4,6) G(2,3,5,7)+2 G(1,6) G(4,8) G(2,3,5,7)+G(1,4,6,8) G(2,3,5,7)+2 G(1,7) G(4,6) G(2,3,5,8)-2 G(1,6) G(4,7) G(2,3,5,8)-G(1,4,6,7) G(2,3,5,8)-2 G(1,5) G(4,8) G(2,3,6,7)-G(1,4,5,8) G(2,3,6,7)+2 G(1,5) G(4,7) G(2,3,6,8)+G(1,4,5,7) G(2,3,6,8)-2 G(1,5) G(4,6) G(2,3,7,8)-G(1,4,5,6) G(2,3,7,8)-2 G(1,8) G(3,7) G(2,4,5,6)+2 G(1,7) G(3,8) G(2,4,5,6)+G(1,3,7,8) G(2,4,5,6)+2 G(1,8) G(3,6) G(2,4,5,7)-2 G(1,6) G(3,8) G(2,4,5,7)-G(1,3,6,8) G(2,4,5,7)-2 G(1,7) G(3,6) G(2,4,5,8)+2 G(1,6) G(3,7) G(2,4,5,8)+G(1,3,6,7) G(2,4,5,8)-2 G(1,8) G(3,5) G(2,4,6,7)+2 G(1,5) G(3,8) G(2,4,6,7)+G(1,3,5,8) G(2,4,6,7)+2 G(1,7) G(3,5) G(2,4,6,8)-2 G(1,5) G(3,7) G(2,4,6,8)-G(1,3,5,7) G(2,4,6,8)-2 G(1,6) G(3,5) G(2,4,7,8)+2 G(1,5) G(3,6) G(2,4,7,8)+G(1,3,5,6) G(2,4,7,8)+2 G(1,8) G(2,7) G(3,4,5,6)-2 G(1,7) G(2,8) G(3,4,5,6)-G(1,2,7,8) G(3,4,5,6)-2 G(1,8) G(2,6) G(3,4,5,7)+2 G(1,6) G(2,8) G(3,4,5,7)+G(1,2,6,8) G(3,4,5,7)+2 G(1,7) G(2,6) G(3,4,5,8)-2 G(1,6) G(2,7) G(3,4,5,8)-G(1,2,6,7) G(3,4,5,8)+2 G(1,8) G(2,5) G(3,4,6,7)-2 G(1,5) G(2,8) G(3,4,6,7)-G(1,2,5,8) G(3,4,6,7)-2 G(1,7) G(2,5) G(3,4,6,8)+2 G(1,5) G(2,7) G(3,4,6,8)+G(1,2,5,7) G(3,4,6,8)+2 G(1,6) G(2,5) G(3,4,7,8)-2 G(1,5) G(2,6) G(3,4,7,8)-G(1,2,5,6) G(3,4,7,8)+G(4,8) G(1,2,3,5,6,7)-G(4,7) G(1,2,3,5,6,8)+G(4,6) G(1,2,3,5,7,8)-G(3,8) G(1,2,4,5,6,7)+G(3,7) G(1,2,4,5,6,8)-G(3,6) G(1,2,4,5,7,8)+G(3,5) G(1,2,4,6,7,8)+G(2,8) G(1,3,4,5,6,7)-G(2,7) G(1,3,4,5,6,8)+G(2,6) G(1,3,4,5,7,8)-G(2,5) G(1,3,4,6,7,8)-G(1,8) G(2,3,4,5,6,7)+G(1,7) G(2,3,4,5,6,8)-G(1,6) G(2,3,4,5,7,8)+G(1,5) G(2,3,4,6,7,8)+G(1,2,3,4,5,6,7,8)$$