Codeforces 438E the "child" and Binary tree NTT Template generation function Application __codeforces

Ask how many binary trees make each of the point weights in the given set C and the weight values and

An FS f_s represents the number of qualified two-forked trees with a total weight of s.
So there
F0=1fs=∑w∈c∑ifixfs−w−i \begin{align} &f_0=1\\ &f_s=\sum_{w\in c}\sum_i f_i\times f_{s-w-i} \end{align}
Found the right and convolution a bit like.
So make
Gk=∑ifixfk−i g_k=\sum_i F_i\times F_{k-i}
Since G (k) =f (k) ⊗f (k) g (k) =f (k) \otimes f (k), that is, the sequence G G is the self convolution of the sequence F F (own disorderly name do not care).
It is known from convolution law that the relationship between the generating function of the sequence G and F F is: g (x) =f2 (x) g (x) =f^2 (x).

There are also fs=∑w∈cgs−w f_s=\sum_{w\in C} g_{s-w}
If the new sequence of G G to move the W bit to the right of the sequence is HW h_w, then there is f