How is recursion a "lost art"? It's just a technique that people use sometimes and nothing too special. Sometimes you get some elegant code with it but it's a tradeoff as it's slower and can cause stackoverflows compared to an iterative alternative that wouldn't.
1! =1, 0! = 1
n! =
n * (
n  1)!
So:
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int factorial(int n)
{
if (n <= 1) return 1;
else return n * factorial(n  1);
}
 
Which is a slower and less stable version of:
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int factorial(int n)
{
int result = 1;
while (n >= 1) {
result *= n;
n;
}
return result;
}
 
I imagine your book probably covers as much detail if not less than the wikipedia article on recursion.
https://en.wikipedia.org/wiki/Recursion_(computer_science)
Read through that if you want to gussyup.
There's no special trick to recursion, if you feel like there's more to it then you're right, but it's mainly specific to different situations so you just have to think logically about it.
On the subject of recursion, as it lends itself towards it (although is not bound by it), I could mention memoization. This is a neat technique that basically involves remembering a result in a subproblem (i.e. storing the result of a toplevel recursion step so that it is only fullycalled once and other times it just looks up the result). But this could be used in any situation where you split the issue up into subproblems.
Algorithms that split up an issue into subproblems come of two main forms: divide and conquer, and dynamic programming.
In dynamic programming one splits a problem into subproblems that don't partition the problem but are all needed to solve the overall problem (i.e. the subproblems overlap, or are repeated). If the subproblems are repeated as a result one can use memoization to cut down on the processing needed to solve the problem, as it will solve a subproblem only once and then if the answer is needed again it can look it up.
In divide and conquer one partitions the input into smaller groups in order to solve smaller problems separately to get the answer. So divide and conquer can be programmed with recursion (but not memoization). An interesting divideandconquer algorithm that can be programmed recursively is Quick Sort.
Further reading:
https://en.wikipedia.org/wiki/Memoization
https://en.wikipedia.org/wiki/Dynamic_programming
https://en.wikipedia.org/wiki/Divide_and_conquer_algorithm
https://en.wikipedia.org/wiki/Quick_sort