I just got some moments of a random variable that have the form as follows.
$E(X^n)=\frac{n!}{(2n-1)!!}e^{n(n+1)x}$
It looks like a log-normal distribution, but not a log-normal distribution. Is there a well-known probability distribution in which moments have the mentioned form?
Too long for a comment:
When a lognormal variable is defined as $$ Y=e^{\mu+\sigma Z}\,,\quad Z\sim N(0,1) $$ then the moments of $Y$ are -as we know- $$ \mathbb E[Y^n]=e^{\mu n +\frac{1}{2}n^2\sigma^2}\,. $$ Therefore, with $\mu=x$ and $\sigma^2=2x$ we get $$ \mathbb E[Y^n]=e^{n(n+n)x}\,. $$ Writing $$ (2n-1)!!=\frac{(2n)!}{2^nn!} $$ we get $$ \frac{n!}{(2n-1)!!}=\frac{2^n(n!)^2}{(2n)!}=\frac{2^n}{2n\choose{n}}\approx\frac{n!}{n^n} $$ where we used that, for large $n$, $$ {2n\choose n}\approx\frac{(2n)^n}{n!} $$ holds.
