Investigating further questions around this question: Example of sequence of graphs which satisfy the Riemann hypothesis? leads to the partition function $Z$ of the Ising model of the graph defined here: Why is this bipartite graph a partial cube, if it is?

It seems that:

$$Z_{n,\beta} = \sum_{k=0}^N c_{N,k} \cdot \exp(-\beta)^{2(N-2k)}$$

where $N = \sum_{i=1}^{n-1} \omega(i) = |E_{n-1}|$ and the coefficients $c_{N,k}$ are somehow mysterious to me at the moment.

Here are some empirical data, where $t = \exp(-\beta)$:

```
n N Z_n
1 0 2
2 0 4
3 1 4*t^2 + 4/t^2
4 2 4*t^4 + 4/t^4 + 8
5 3 4*t^6 + 12*t^2 + 12/t^2 + 4/t^6
6 4 4*t^8 + 16*t^4 + 16/t^4 + 4/t^8 + 24
7 6 4*t^12 + 8*t^8 + 28*t^4 + 28/t^4 + 8/t^8 + 4/t^12 + 48
8 7 4*t^14 + 12*t^10 + 36*t^6 + 76*t^2 + 76/t^2 + 36/t^6 + 12/t^10 + 4/t^14
9 8 4*t^16 + 16*t^12 + 48*t^8 + 112*t^4 + 112/t^4 + 48/t^8 + 16/t^12 + 4/t^16 + 152
```

**Question: What are the coefficients $c_{N,k}$?**

I am not asking for a proof, just some empirical direction, if it is possible and not so difficult to do.

Here is the python source code for further computations, if needed.

If it is of interest to the question:

The Hamiltonian function is maximal possible:

$$H_n = - \sum_{(u,v) \in E_n} \sigma_u \sigma_v = |E_n| = \sum_{k=1}^n \omega(n) \approx n(\log(\log(n))+B_1)+o(n)$$

where $B_1 \approx 0.26$ is Mertens constant.