zscoder introduced Generating functions in Competitive Programming in his blog, which give a method to get the formula of general term of a array by it's generating function.

But sometime the generating function is somewhat complicated, for instance, the array http://oeis.org/A054726 which has generating function:

and we can calculate the first n term by poly sqrt with fft in $$$O(n \log n)$$$

But this array have a recurrence formula:

which will be much easy to calculate.

My question is how can I get recurrence formula by it's generating function in general(or some special form) ?

I know how to do this now, thanks qwaszx who give me a hint.

If $$$f(z) = p(z)^{ \frac{1}{m} }$$$ is generating function of $$$a_n$$$, where $$$p(z)$$$ is a polynomial. then

compare the coefficient of $$$z^n$$$， and we will get the recurrence formula of $$$a_n$$$

Let us calculate the above example: $$$f(z) = 1 + \frac{3}{2} z - z^2 - \frac{z}{2} \sqrt{1 - 12 z + 4 z^2}$$$

let $$$b_n = a_{n+1}$$$ and $$$g(z)$$$ is generating function of $$$b_n$$$, then

so we get differential of $$$g(z)$$$

and

compare the coefficient of $$$z^n$$$, we have $$$(n + 1)b_{n+1} - 12 n b_n + 4(n-1)b_{n-1} = 4 b_{n-1} - 6 b_n$$$, so

so we archive our goal: $$$a_{n} = \frac{ (12 n - 30) a_{n-1} - (4 n - 16) a_{n-2} }{n-1}$$$