I am trying to compute the following quantity: $$ g_n(x) = \sum_{\lambda \vdash n} \prod_{h \in \mathcal{H}(\lambda)} \frac{1}{h^2} \exp\left[x\sum_i \binom{\lambda_i}{2} - \binom{\lambda_i'}{2}\right] $$ where $\lambda$ is an integer partition of $n$ and $\mathcal{H}(\lambda)$ represents the set of hook length in the partition $\lambda$. The $\lambda_i$, with $\lambda_1\geq \lambda_2 \geq \ldots$ are the parts of $\lambda$. While $\lambda'_j$ are the parts of the dual partition, i.e. $$ \lambda_j' = \# \{\lambda_i | \lambda_i \geq j \} $$
If I expand in powers of $x$, I get $$ g_n(x) = \sum_{k=0}^\infty \frac{g_{n,k} x^k}{k!} $$ where the coefficients are expressed as $$ g_{n,k} = \sum_{\lambda \vdash n} \prod_{h \in \mathcal{H}(\lambda)} \frac{1}{h^2} \left[\sum_i \binom{\lambda_i}{2} - \binom{\lambda_i'}{2}\right]^k $$ By direct inspection, I can see that the coefficients behave as $$ g_{n,k} = \frac{1}{2}\frac{1}{n!} (n-1) p_k(n) $$ where $p_k(n)$ is a polynomial of degree $2k - 1$, vanishing for all odd $k$'s. I am interested in evaluating the sequence $p_k(n=1)$. For the first few, I get the integer values, ($m = 1,2,\ldots$) $$ p_{2m}(n=1) = \{1, -4, 146, -20194, 7104506, -5064157834, 6331350708866, -12564039882409474, 36735333764214529226, -149114709095955067405114, \ldots $$
I do not know which methods to use to get an analytic expression for these polynomials in order to be able to compute the $n=1$ behavior. Any suggestions would be very helpful!
