As promised, I've upgraded my last question.
Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, start tiling $Y_{n,k}$ using monomers ($1\times1$ squares) and dimers ($1\times2$ and $2\times1$ rectangles).
Next, insert the hook-lengths $h(\square)$ into each cell $\square\in Y_{n,k}$. Associate a weight: a monomer at $\square$ receives $h(\square)$, a dimer sitting on $\square$ and $\square'$ gets the product $h(\square)\cdot h(\square')$. Each tiling $T$ will have weight assigned as the sum of the weights of its monomers and dimers. Let $b_{n,k}$ be the entire sum of the weights of all possible tiltings of $Y_{n,k}$. For example, if $n=3, k=1$ then we get $$b_{3,1}=(3+2+1)+(3\cdot2+1)+(3+2\cdot1)=6+7+5=18.$$ The first few values are: $b_1=1, b_2=5, b_3=18, b_4=59, b_5=162$.
QUESTION. Are these generating function $G_k(x)=\sum_nb_{n,k}x^n$ rational functions? Or, can you verify this for small $k$, such as $k=2,3,4$, etc?
Remark. Fedor's comment led to $G_1(x)=\frac{x(1+x+5x^3-3x^4)}{(1-x-x^2)^4}$. A rational function.
