Let $\mathbb{Y}_n$ denote the set of all partitions of $n\in\mathbb{N}$ and $\mathbb{Y}$ Young's lattice of all partitions. The partition function $g_0(n)=\sum_{\lambda\in\mathbb{Y}_n}1$ has an asymptotic formula proven by Hardy and Ramanujan. This sequence is A000041 in the OEIS.

Given a partition $\lambda$, let $\dim\lambda$ be the number of paths in $\mathbb{y}$ starting at $\emptyset$ and ending at $\lambda$, also known as the number of standard Young tableau of shape $\lambda$. The number of Young tableau of size $n$ is given by $g_1(n)=\sum_{\lambda\in\mathbb{Y}_n}\dim\lambda$ which has known asymptotics. Incidentally, $g_1(n)$ counts the number of involutions in $\mathfrak{S}_n$. This is sequence A000085 in the OEIS.

Basic representation theory of $\mathfrak{S}_n$ and Stirling's approximation give $$g_2(n)=\sum_{\lambda\in\mathbb{Y}_n}(\dim\lambda)^2=n!\sim \sqrt{2\pi n}\,n^ne^{-n}.$$ A special feature of $g_2(n)$ is: it allows an analytic continuation via the Euler's Gamma function $\Gamma(n)$.

Define $g_k(n)=\sum_{\lambda\in\mathbb{Y}_n}(\dim\lambda)^k$. Now, I would like to inquire:

Question 1.What is the growth rate of $g_3(n)$?

Question 2.Is there an analytic continuation of $g_3(n)$, as for $g_2(n)$, in the variable $n$?