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
Your particular sequence
falls into the group covered by the algorithm of Chapter 8.