Finding the summation formula for the recurrence relation $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$ The exponential generating function of this recurrence relation, $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$, is
$$f(x)=e^{x + \frac{x^2}{2}}$$
Multiplying the exponential generating functions for each term, we get,
$$=\ \sum_{j\geq0}\;\frac{1}{j!}x^j\cdot\sum_{k\geq0}\;\frac{1}{2^kk!}x^{2k}$$
$$=\ \sum_{j,\ \ k\geq0}\;\frac{1}{j!}\cdot\frac{1}{2^kk!}x^{j+2k}$$
Taking $n\ =\ j\ +\ 2k$,
$$=\ \sum_{n-2k,\ \ k\geq0}\;\frac{1}{(n-2k)!}\cdot\frac{1}{2^kk!}x^n$$
What steps can I take from here to find the summation formula for the recurrence relation, i.e.,
$$\;\sum_{n\ge0}\Big(\sum_{k=0}^{[n/2]}\frac{n!}{(n-2k)!2^kk!}\Big)\frac{x^n}{n!}.$$
 A: *

*Your latter sum,
$$\sum_{n=0}^\infty a_n\frac{x^n}{n!},$$
is just $f(x)=e^{x+x^2/2}$, as shown in your own post, where
$$a_n:=T_n=\sum_{k=0}^{[n/2]}\frac{n!}{(n-2k)!2^k k!}=\sum_{k=0}^\infty\binom n{2k}\frac{(2k)!}{2^k k!}.\tag{1}$$


*A closed-form expression for $a_n$ apparently does not exist. Anyway, Mathematica does not know it:



*However, one may note that your function $f$ is the moment generating function of the normal distribution with mean $1$ and variance $1$. So, if $Z$ is a standard normal random variable, then
$$a_n=E(Z+1)^n=\sum_{j=0}^n\binom nj EZ^j=\sum_{0\le k\le n/2}\binom n{2k} EZ^{2k},$$
whence (1) follows. The recurrence relation $a_n=(n-1)a_{n-2}+a_{n-1}$ also follows, by writing
$$E(Z+1)^n=EZ(Z+1)^{n-1}+E(Z+1)^{n-1}$$
and taking the first of the latter two integrals by parts.

Mathematica can only give the trivial answer for $E(Z+1)^n$:

A: The probabilist's Hermite polynomials (OEIS A066325) $He_n(x)$ have the e.g.f.
$$e^{He.(t)x} = e^{xt} e^{-\frac{x^2}{2}} = e^{xt} e^{h.x} = e^{(h.+ t)x}$$
where
$$He_n(t) = (h. + t)^n =\sum_{k=0}^n \binom{n}{k} h_k t^{n-k}$$
with
$$h_n = \cos(\frac{\pi n}{2}) \frac{n!}{(n/2)!} 2^{-n/2}.$$
Consequently,
$$f(x) = e^{a.x} = e^{x+ x^2/2}$$
$$= e^{-He(i)ix},$$
so
$$a_n = (-i)^n He_n(i) = \hat{He}_n(1),$$
where $\hat{H}_n(x)$ are the row polynomials of A099174 with row sums, $a_n$, the sequence OEIS A000085 with extensive characterizations.
