Partial sum of Weingarten functions over symmetric group

I have a question about partial sums of Weingarten functions. The Weingarten functions are defined as $$ E_U[U_{i_1,j_1}\dotsm U_{i_k,j_k}U^*_{i'_1,j'_1}\dotsm U^*_{i'_k,j'_k}]=\sum_{\alpha,\beta \in \mathbb{S}_k} [i_1=i'_{\alpha(1)},\dotsc,i_k=i'_{\alpha(k)}]\cdot [j_1=j'_{\alpha(1)},\dotsc,j_k=j'_{\alpha(k)}]\cdot W(\alpha \beta^{-1},d), $$ where the expectation is over $d\times d$ Haar random matrices. Now a well known property about these functions are $\sum_{\sigma\in \mathbb{S}_k}W(\sigma,d)=d!/(d+k)!$. Do we know what happens to the partial sum $$\sum_{\sigma\in \mathbb{S}_{k/2}\times \mathbb{S}_{k/2}}W(\sigma,d)$$ (and similarly if we consider $\sum_{\sigma\in \mathbb{S}_{k/3}\times \mathbb{S}_{k/3}\times \mathbb{S}_{k/3}}W(\sigma,d)$ and so on)? Are there closed form expressions for these quantities? References would be helpful .