Residual Polynomial

寫出所有的$f_i(x)$出來，$f_{i,j}$表示$f_i(x)$的第$j$項系數

$$?\begin{array}{cccccc}{f}_{1,0}& f_{1,1}& f_{1,2}& \dots & f_{1,n?1}& f_{1,n}\\ f_{2,0}& f_{2,1}& f_{2,2}& \dots & f_{2,n?1}& f_{2,n}\\ f_{3,0}& f_{3,1}& f_{3,2}& \dots & f_{3,n?1}& f_{3,n}\\ ?& ?& ?& ?& ?& ?\\ f_{n?1,0}& f_{n?1,1}& f_{n?1,2}& \dots & f_{n?1,n?1}& f_{n?1,n}\\ f_{n,0}& f_{n,1}& f_{n,2}& \dots & f_{n,n?1}& f_{n,n}\end{array}?$$
第一行是$a_0,a_1,a_2,\dots ,a_{n?1},a_n$。

考慮下一行如何從上一行轉移，分兩種：

一、$f_{i,j}\times c_{i+1}=f_{i+1,j}$

二、$f_{i,j}\times j\times b_{i+1}=f_{i+1,j?1}$

可以想想成每個點有一條向下的連邊$value=c$，同時有一條向左下的連邊$value=b\times j$。

我們從$f_{1,i}$出發，走向$f_{n,j}$，顯然選擇了$i?j$個$b\times j$，所以還有$n?1?(i?j)$個$c$。

先不考慮$j$，只算$b,c$的貢獻，設選$b$的生成函數為$F(x)$，對于某個點有$F(x)=c+bx$，因為上面的轉移是具有區間性的。

所以$F(l,r,x)=F(l,mid,x)?F(mid,r,x)$，也就是在左區間選幾個，右區間選幾個，然后組合一下，這個組合剛好滿足卷積的形式。

這一步就可以遞歸$O(n\log n\log n)$求解了。

接下來考慮$j$的影響：

$f_{n,k}=\sum _{i?k=j}F(j)f_{1,i}\frac{i!}{k!}$

這里可以簡單理解一下，每次只能向左向下移動，k是我們最后在的列，只能從初始列$i\ge k$的列轉移過來。

$\left(k!\times f_{n,k}\right)=\sum _{i?k=j}F(j)\left(f_{1,i}\times i!\right)$

$設H(k)=k!\times f_{n,k},G(i)=f_{1,i}\times i!$，$考慮把G翻轉，G(n?i)=G(i)$，

$H(k)={\sum }_{i+j=n?k}F(j)G(i)$，于是再做一次卷積即可求得答案。

#include <bits/stdc++.h>using namespace std;const int mod = 998244353, inv2 = mod + 1 >> 1;namespace Quadratic_residue {struct Complex {int r, i;Complex(int _r = 0, int _i = 0) : r(_r), i(_i) {}};int I2;Complex operator * (const Complex &a, Complex &b) {return Complex((1ll * a.r * b.r % mod + 1ll * a.i * b.i % mod * I2 % mod) % mod, (1ll * a.r * b.i % mod + 1ll * a.i * b.r % mod) % mod);}Complex quick_pow(Complex a, int n) {Complex ans = Complex(1, 0);while (n) {if (n & 1) {ans = ans * a;}a = a * a;n >>= 1;}return ans;}int get_residue(int n) {mt19937 e(233);if (n == 0) {return 0;}if(quick_pow(n, (mod - 1) >> 1).r == mod - 1) {return -1;}uniform_int_distribution<int> r(0, mod - 1);int a = r(e);while(quick_pow((1ll * a * a % mod - n + mod) % mod, (mod - 1) >> 1).r == 1) {a = r(e);}I2 = (1ll * a * a % mod - n + mod) % mod;int x = quick_pow(Complex(a, 1), (mod + 1) >> 1).r, y = mod - x;if(x > y) swap(x, y);return x;} }const int N = 1e6 + 10;int r[N], inv[N], b[N], c[N], d[N], e[N], t[N];int quick_pow(int a, int n) {int ans = 1;while (n) {if (n & 1) {ans = 1ll * a * ans % mod;}a = 1ll * a * a % mod;n >>= 1;}return ans; }void get_r(int lim) {for (int i = 0; i < lim; i++) {r[i] = (i & 1) * (lim >> 1) + (r[i >> 1] >> 1);} }void get_inv(int n) {inv[1] = 1;for (int i = 2; i <= n; i++) {inv[i] = 1ll * (mod - mod / i) * inv[mod % i] % mod;} }void NTT(int *f, int lim, int rev) {for (int i = 0; i < lim; i++) {if (i < r[i]) {swap(f[i], f[r[i]]);}}for (int mid = 1; mid < lim; mid <<= 1) {int wn = quick_pow(3, (mod - 1) / (mid << 1));for (int len = mid << 1, cur = 0; cur < lim; cur += len) {int w = 1;for (int k = 0; k < mid; k++, w = 1ll * w * wn % mod) {int x = f[cur + k], y = 1ll * w * f[cur + mid + k] % mod;f[cur + k] = (x + y) % mod, f[cur + mid + k] = (x - y + mod) % mod;}}}if (rev == -1) {int inv = quick_pow(lim, mod - 2);reverse(f + 1, f + lim);for (int i = 0; i < lim; i++) {f[i] = 1ll * f[i] * inv % mod;}} }void polyinv(int *f, int *g, int n) {if (n == 1) {g[0] = quick_pow(f[0], mod - 2);return ;}polyinv(f, g, n + 1 >> 1);for (int i = 0; i < n; i++) {t[i] = f[i];}int lim = 1;while (lim < 2 * n) {lim <<= 1;}get_r(lim);NTT(t, lim, 1);NTT(g, lim, 1);for (int i = 0; i < lim; i++) {int cur = (2 - 1ll * g[i] * t[i] % mod + mod) % mod;g[i] = 1ll * g[i] * cur % mod;t[i] = 0;}NTT(g, lim, -1);for (int i = n; i < lim; i++) {g[i] = 0;} }void polysqrt(int *f, int *g, int n) {if (n == 1) {g[0] = Quadratic_residue::get_residue(f[0]);return ;}polysqrt(f, g, n + 1 >> 1);polyinv(g, b, n);int lim = 1;while (lim < 2 * n) {lim <<= 1;}get_r(lim);for (int i = 0; i < n; i++) {t[i] = f[i];}NTT(g, lim, 1);NTT(b, lim, 1);NTT(t, lim, 1);for (int i = 0; i < lim; i++) {g[i] = (1ll * inv2 * g[i] % mod + 1ll * inv2 * b[i] % mod * t[i] % mod) % mod;b[i] = t[i] = 0;}NTT(g, lim, -1);for (int i = n; i < lim; i++) {g[i] = 0;} }void derivative(int *a, int *b, int n) {for (int i = 0; i < n; i++) {b[i] = 1ll * a[i + 1] * (i + 1) % mod;} }void integrate(int *a, int n) {for (int i = n - 1; i >= 1; i--) {a[i] = 1ll * a[i - 1] * inv[i] % mod;}a[0] = 0; }void polyln(int *f, int *g, int n) {polyinv(f, b, n);derivative(f, g, n);int lim = 1;while (lim < 2 * n) {lim <<= 1;}get_r(lim);NTT(g, lim, 1);NTT(b, lim, 1);for (int i = 0; i < lim; i++) {g[i] = 1ll * g[i] * b[i] % mod;b[i] = 0;}NTT(g, lim, -1);for (int i = n; i < lim; i++) {g[i] = 0;}integrate(g, n); }void polyexp(int *f, int *g, int n) {if (n == 1) {g[0] = 1;return ;}polyexp(f, g, n + 1 >> 1);int lim = 1;while (lim < 2 * n) {lim <<= 1;}polyln(g, d, n);for (int i = 0; i < n; i++) {t[i] = (f[i] - d[i] + mod) % mod;}t[0] = (t[0] + 1) % mod;get_r(lim);NTT(g, lim, 1);NTT(t, lim, 1);for (int i = 0; i < lim; i++) {g[i] = 1ll * g[i] * t[i] % mod;t[i] = d[i] = 0;}NTT(g, lim, -1);for (int i = n; i < lim; i++) {g[i] = 0;} }/*b存放多項式逆，c存放多項式開根，d存放多項式對數ln，e存放多項式指數exp，t作為中間轉移數組,如果要用到polyln，得提前調用get_inv(n)先預先得到我們想要得到的逆元范圍。 */int B[N], C[N], f[N], fac[N], inv1[N], n;vector<int> F[N];int A1[N], B1[N];void init() {fac[0] = 1;for (int i = 1; i < N; i++) {fac[i] = 1ll * fac[i - 1] * i % mod;}inv1[N - 1] = quick_pow(fac[N - 1], mod - 2);for (int i = N - 2; i >= 0; i--) {inv1[i] = 1ll * inv1[i + 1] * (i + 1) % mod;} }void merge(int rt) {int ls = rt << 1, rs = rt << 1 | 1;int n = F[ls].size(), m = F[rs].size();F[rt].resize(n + m);for (int i = 0; i < n + m; i++) {F[rt][i] = 0;}if (n <= 50 && m <= 50) {for (int i = 0; i < n; i++) {for (int j = 0; j < m; j++) {F[rt][i + j] = (F[rt][i + j] + 1ll * F[ls][i] * F[rs][j] % mod) % mod;}}return ;}for (int i = 0; i < n; i++) {A1[i] = F[ls][i];}for (int i = 0; i < m; i++) {B1[i] = F[rs][i];}int lim = 1;while (lim < n + m) {lim <<= 1;}get_r(lim);NTT(A1, lim, 1);NTT(B1, lim, 1);for (int i = 0; i < lim; i++) {A1[i] = 1ll * A1[i] * B1[i] % mod;}NTT(A1, lim, -1);for (int i = 0; i < n + m; i++) {F[rt][i] = A1[i];}for (int i = 0; i < lim; i++) {A1[i] = B1[i] = 0;} }void divide(int rt, int l, int r) {if (l == r) {F[rt].push_back(C[l]);F[rt].push_back(B[l]);return ;}int mid = l + r >> 1;divide(rt << 1, l, mid), divide(rt << 1 | 1, mid + 1, r);merge(rt);F[rt << 1].clear(), F[rt << 1 | 1].clear(); }int main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);int T;init();scanf("%d", &T);while (T--) {scanf("%d", &n);for (int i = 0; i <= n; i++) {scanf("%d", &f[i]);f[i] = 1ll * f[i] * fac[i] % mod;}for (int i = 0; i <= n - 2; i++) {scanf("%d", &B[i]);}for (int i = 0; i <= n - 2; i++) {scanf("%d", &C[i]);}divide(1, 0, n - 2);int lim = 1;while (lim <= 2 * n) {lim <<= 1;}get_r(lim);for (int i = 0; i <= n; i++) {A1[i] = f[n - i];}for (int i = 0; i < F[1].size(); i++) {B1[i] = F[1][i];}F[1].clear();NTT(A1, lim, 1);NTT(B1, lim, 1);for (int i = 0; i < lim; i++) {A1[i] = 1ll * A1[i] * B1[i] % mod;}NTT(A1, lim, -1);for (int i = 0; i <= n; i++) {printf("%lld%c", 1ll * inv1[i] * A1[n - i] % mod, i == n ? '\n' : ' ');}for (int i = 0; i < lim; i++) {A1[i] = B1[i] = 0;}}return 0; }

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