P5641 【CSGRound2】开拓者的卓识

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求$sum_{k,1,1},sum_{k,1,2}….sum_{k,1,n}$

从上向下拓展的想$[i,i]->[i-a,i+b]->[1,r]$，经过$k-1$步走到了$[1,r]$，考虑走法就是$[1,i]$选$k-1$个$\times[i,r]$选$k-1$个。根据组合知识从$n$个东西里面选$m$个，可重复$={n+m-1\choose n-1},{n+m-1\choose m}$

$NTT$递推预处理下组合数。

代码 
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#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define pii pair<int, int>
#define mk make_pair
const int N = 1e6 + 10;

const int mod = 998244353;
const int P = 998244353, gi = 3;
const double pi = acos(-1.0);
int inc(int x, int y, int mo)
{
    if (x + y >= mo)
        return (x + y - mo);
    else
        return (x + y);
}

int del(int x, int y, int mo)
{
    if (x - y < 0)
        return (x - y + mo);
    else
        return (x - y);
}
int read()
{
    int x = 0, f = 1;
    char c = getchar();
    while (c < '0' || c > '9')
    {
        if (c == '-')
            f = -1;
        c = getchar();
    }
    while (c >= '0' && c <= '9')
        x = (x << 1) + (x << 3) + c - '0', c = getchar();
    return x * f;
}
int qpow(int a, int x, int mo)
{
    int res = 1;
    while (x)
    {
        if (x & 1)
            res = 1ll * res * a % mo;
        x >>= 1;
        a = 1ll * a * a % mo;
    }
    return res;
}
int rev[N];
void NTT(int *A, int n, int inv)
{
    for (int i = 0; i < n; i++)
        if (i < rev[i])
            swap(A[i], A[rev[i]]);
    for (int l = 1; l < n; l <<= 1)
    {
        int tt = qpow(gi, (P - 1) / (l << 1), P);
        int temp = (inv == 1 ? tt : qpow(tt, P - 2, P));
        for (int i = 0; i < n; i += (l << 1))
        {
            int omega = 1;
            for (int j = 0; j < l; j++, omega = 1ll * omega * temp % P)
            {
                int x = A[i + j], y = 1ll * omega * A[i + j + l] % P;
                A[i + j] = inc(x, y, P);
                A[i + j + l] = del(x, y, P);
            }
        }
    }
    int invv = qpow(n, P - 2, P);
    if (inv == -1)
        for (int i = 0; i < n; i++)
            A[i] = 1ll * A[i] * invv % P;
}

void NTTX(int *a, int n, int *b, int m, int *ans)
{
    int ML = 1, bit = 0;
    while (ML < n + m)
        ML <<= 1, bit++;
    for (int i = 0; i < ML; i++)
        rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (bit - 1));
    NTT(a, ML, 1);
    NTT(b, ML, 1);
    for (int i = 0; i < ML; i++)
        ans[i] = 1ll * a[i] * b[i] % P;
    NTT(ans, ML, -1);
}
int f[N], g[N], a[N], inv[N], c[N];
int main()
{
    int n = read(), k = read();
    for (int i = 1; i <= n; i++)
        a[i] = read();
    inv[0] = inv[1] = 1;
    for (int i = 2; i <= n; i++)
        inv[i] = 1ll * (mod - mod / i) * inv[mod % i] % mod;
    g[1] = 1;
    for (int i = 2; i <= n; i++)
    {
        g[i] = 1ll * (i + k - 2) % mod * g[i - 1] % mod * inv[i - 1] % mod;
        //cout << g[i] << " ";
    }

    for (int i = 1; i <= n; i++)
        g[i] = 1ll * g[i] * a[i] % mod;
    g[0] = 0;
    f[0] = 1;
    for (int i = 1; i <= n; i++)
    {
        f[i] = 1ll * (i + k - 1) % mod * f[i - 1] % mod * inv[i] % mod;
        //cout << i + k - 1 << " " << inv[i] << " " << f[i] << " " << endl;
        ;
    }

    NTTX(f, n, g, n, c);
    for (int i = 1; i <= n; i++)
        printf("%d ", c[i]);
}