Let $\mathfrak{S}\_n$ denote the permutation group, and $I\_0(n)=\sum\_{j\geq0}\binom{n}{2j}\frac{(2j)!}{2^jj!}$ stand for involutions [see A000085 for more interpretations](https://oeis.org/A000085). There is also these numbers $I\_1(n)=\sum\_{j\geq0}\binom{n}jI\_0(j)I\_0(n-j)$ described [in A000898 by several means](https://oeis.org/A000898).


Let me introduce the numbers $I\_2(n)=\sum\_{j\geq0}\binom{n}jI\_0(j)^2I\_0(n-j)^2$. I was able to verify the exponential generating function
$$\sum\_{n\geq0}I\_2(n)\frac{x^n}{n!}=\frac1{1-x^2}e^{\frac{2x}{1-x}}.$$
However, it is desirable to know:



> 
> **Question.** Is there a combinatorial meaning to the numbers $I\_2(n)$?
> 
> 
> 


**Remark.** Of course, it is also interesting if one can provide any other context where $I\_2(n)$ appears.



Provoking involutions further
The generating function for involutions with respect to the number of fixed points is given by an evaluation of  [Hermite polynomials](https://en.wikipedia.org/wiki/Hermite_polynomials) . The bilinear generating function of Hermite polynomials is given by ["Mehler's formula"](https://en.wikipedia.org/wiki/Mehler_kernel): 
$$\sum\_{n\geq 0}\frac{t^n H\_n(x)H\_n(y)}{2^nn!}=\frac{1}{\sqrt{1-t^2}}\exp\left(\frac{t^2(x^2+y^2)-2txy}{t^2-1}\right).$$


Foata wrote the paper ["A combinatorial proof of the Mehler formula"](http://www.sciencedirect.com/science/article/pii/0097316578900663) where he gives a combinatorial interpretation and bijective proof of the formula. Mehler's formula (after an appropriate specialization) is technically related to the square root of your generating function. Your generating function also has a similar refinement and it's what Zeilberger calls ["the heterosexual version of Mehler's formula"](https://arxiv.org/abs/math/9807074). To see your identity in Zeilberger's set up you need to set $x=y=1$ and $s=t$ in his main identity.


