Chapter 8 of Petkovsek--Wilf--Zeilberger $A=B$ starts as follows:

"If you want to evaluate a given sum in closed form, so far the tools that have been
described in this book have enabled you to find a recurrence relation with polynomial
coefficients that your sum satisfies. If that recurrence is of order 1 then you are finished;
you have found the desired closed form for your sum, as a single hypergeometric
term. If, on the other hand, the recurrence is of order $\ge2$ then there is more work
to do. How can we recognize when such a recurrence has hypergeometric solutions,
and how can we find all of them?"

"In this chapter we discuss the question of how to recognize when a given recurrence
relation with polynomial coefficients has a closed form solution. We first take the
opportunity to define the term *closed form*."

"A function $f(n)$ is said to be of closed form if it is equal to a linear
combination of a fixed number, $r$, say, of hypergeometric terms. The number $r$ must
be an absolute constant, i.e., it must be independent of all variables and parameters
of the problem."

"Take a definite sum of the form $f(n) = \sum_k F(n; k)$ where the summand $F(n; k)$
is hypergeometric in both its arguments. Does this sum have a closed form? The
material of this chapter, taken together with the algorithm of Chapter 6, provides a
complete algorithmic solution of this problem."

There are many examples in this chapter which illustrate the algorithm, like
Ap\'ery's numbers
$$
\sum_k{\binom{n+k}k}^2{\binom{n}k}^2.
$$
Your particular sequence
$$
\sum_k\frac{n!}{i!(i+1)!(n-2i)!}
$$
falls into the group covered by the algorithm of Chapter 8.