rxd and functions

已知$f(x)= sum_{i=0}^{n} c_{i} {x}^{i}.$
定义变换$Tr(f,a)$，使得$g(x)=Tr(f,a)=f(x-a).$
已知$g_{0}=f$，且有$g_{i}=Tr(g_{i-1}, a_{i}),g_{m} = sum_{i=0}^{n} b_{i} {x}^{i}$，输入$m$。
求$b_{i}$对$998244353$取模。

推式子时间到。

$$f(x) = sum_{i=0}^{n} c_{i} {x}^{i}$$ $$Tr(f, a)=f(x-a)$$ $$g_{0}=f, g_{i}=Tr(g_{i-1}, a_{i}).$$ $$g_{1}= sum_{i=0}^{n} c_{i}{(x-a_{1})}^{i}.$$ $$g_{2}= sum_{i=0}^{n} c_{i} {(x - a_{1} - a_{2})}^{i}.$$ $$...$$ $$g_{m} = sum_{i=0}^{n} c_{i} {(x-a_{1}-a_{2}-a_{3}...)}^{i}.$$ $$S = sum_{i=1}^{m} a_{i}.$$ $$g_{m} = sum_{i=0}^{n} c_{i} {(x-S)}^{i}.$$ $$=b_{i}{x}^{i}.$$ $$b_{i} = sum_{j=i}^{n}c_{j} C_{j}^{i} {(-S)}^{j-i}.$$ $$= sum_{j=i}^{n} c_{j} times frac{j!}{(j-i)!i!} {(-S)}^{j-i}.$$ $$= frac{1}{i!} sum_{j=i}^{n} c_{j} frac{j!}{(j-i)!} {(-S)}^{j-i}.$$ $$= frac{1}{i!} sum_{j=0}^{n-i} c_{i+j} (i+j)! frac{ {(-S)}^{j} }{j!}.$$ $$= frac{1}{i!} sum_{j=0}^{n-i} c_{i+j} (i+j)! frac{ {(-S)}^{n-(n-j)} }{(n-(n-j))!}.$$ $$= frac{1}{i!} times (A*B)_{n+i}.$$

其中$A_{i} = c_{i} times i!,B_{i} = frac{ {(-S)}^{n-i} }{(n-i)!}.$

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131

using namespace std;
typedef long long ll;
#ifndef ONLINE_JUDGE
#define dbg(x...) do{cout << "