Epic Convolution

Epic Convolution！

话说卡老师到底会不会 Epic Convolution II 啊，这玩意有没有被解决啊

由于这题是五合一题 所以代码里面很多都用折纸的天使命名 ×

尝试写一篇人话题解。

在做这道题之前，你需要仔细阅读 具体数学（Concrete Mathematics） 的 6.2 章节，下面列出一些之后要用到的东西：

\[m!\begin{Bmatrix}n\\m\end{Bmatrix}=\sum_k k^n\binom{m}{k}(-1)^{m-k} \]

\[\begin{Bmatrix}n\\m\end{Bmatrix}=\sum_k \frac{k^n(-1)^{m-k}}{k!(m-k)!} \]

\[\left\langle\begin{matrix}n\\m\end{matrix}\right\rangle=\sum_{k}\begin{Bmatrix}n\\k\end{Bmatrix}\binom{n-k}{m}(-1)^{n-k-m}k!=\sum_{k=0}^m\binom{n+1}{k}(m+1-k)^n(-1)^k \]

\[\left\langle\left\langle\begin{matrix}n\\k\end{matrix}\right\rangle\right\rangle=(k+1)\left\langle\left\langle\begin{matrix}n-1\\k\end{matrix}\right\rangle\right\rangle+(2n-1-k)\left\langle\left\langle\begin{matrix}n-1\\k-1\end{matrix}\right\rangle\right\rangle \]

\[\left\langle\left\langle\begin{matrix}n\\k\end{matrix}\right\rangle\right\rangle=\sum_{i=0}^k\binom{2n+1}{i}(-1)^{i}\begin{Bmatrix}n+k+1-i\\k+1-i\end{Bmatrix} \]

接下来依次分析各个 Subtask。

Subtask 1

\[\begin{aligned} f_n&=\sum_{k=0}^nk^ng_kh_{n-k}\\ &=\sum_{k=0}^{m-1}k^n \end{aligned} \]

由于 \(n\) 固定，预处理幂的前缀和即可。

复杂度为 \(O(n)-O(1)\)​。

namespace Shemesh{
	ll ans[N];
    void Metatron(){
        IO>>n>>q;
        rep(i,1,n-1)ans[i]=(ans[i-1]+qpow(i,n))%P;
    }
	void Shemesh(){
		while(q--){
			ll m;
			IO>>m;
			IO<<ans[m-1]<<'\n';
		}
	}
}// Metatron Shemesh 绝灭天使·日轮

Subtask 2

\[f_n=\sum_{k=0}^{n}k^n\frac{1}{(k+m+1)!}\frac{(-1)^{n-k-m}}{(n-k-m)!} \]

显然这里可以提一个组合数，两边乘以 \((n+1)!\) 可得

\[(n+1)!f_n=\sum_{k=0}^{n}k^n\binom{n+1}{k+m+1}(-1)^{n-k-m} \]

然后这个 \(k^n\)​ 显然要拿去配斯特林数，我们观察一下这个式子，然后来尝试强行凑一个。

\[\begin{Bmatrix}n\\m\end{Bmatrix}=\sum_k \frac{k^n(-1)^{m-k}}{k!(m-k)!} \]

那么组合数可以产生求和符号，首先可以把 \((-1)^{n-k-m}\) 拆成 \((-1)^{n-j-m}(-1)^{j-k}\)，\(j\) 是外层求和枚举的变量，然后来考虑分母怎么动，这个时候就可以把斯特林数的式子改一下变成：

\[\begin{Bmatrix}n\\j\end{Bmatrix}=\sum_k \frac{k^n(-1)^{j-k}}{k!(j-k)!} \]

那么我们整一个 \(\dbinom{j}{k}\) 就可以搞出来了，下面考虑怎么把 \(\dbinom{n+1}{k+m+1}\) 搞成求和形式。

事实上根据生成函数容易证明：

\[\binom{n+1}{k+m+1}=\sum_{j}\binom{j}{k}\binom{n-j}{m} \]

然后带进去就可以得到我们想要的式子：

\[\begin{aligned} \sum_{k=0}^{n}k^n\binom{n+1}{k+m+1}(-1)^{n-k-m}&=\sum_{k=0}^{n}k^n(-1)^{n-k-m}\sum_{j}\binom{j}{k}\binom{n-j}{m}\\ &=\sum_{j}(-1)^{n-j-m}\binom{n-j}{m}\sum_{k=0}^{n}k^n(-1)^{j-k}\binom{j}{k}\\ &=\sum_{j}(-1)^{n-j-m}\binom{n-j}{m}j!\begin{Bmatrix}n\\j\end{Bmatrix} \end{aligned} \]

然后你就会发现和欧拉数的通项一完全一致，用通项二带进去可得

\[f_n=\frac{1}{(n+1)!}\sum_{k=0}^m\binom{n+1}{k}(m+1-k)^n(-1)^k \]

预计算幂和阶乘，单次 \(O(m)\)​ 即可。

复杂度 \(O(nm)-O(m)\)。

namespace Malakh{
	ll fac[N],invfac[N],pw[25][N];
	void Metatron(ll n,ll m){
		fac[0]=1;
		rep(i,1,n)fac[i]=fac[i-1]*i%P;
		invfac[n]=qpow(fac[n],P-2);
		Rep(i,n-1,0)invfac[i]=invfac[i+1]*(i+1)%P;
		rep(i,1,m){
			pw[i][0]=1;
			rep(j,1,n)pw[i][j]=pw[i][j-1]*i%P;
		}
	}
	ll C(ll n,ll m){return fac[n]*invfac[m]%P*invfac[n-m]%P;}
	void Malakh(){
		for(IO>>q;q--;){
			IO>>n>>m;
			ll ans=0;
			rep(k,0,m){
				if(k&1^1)ans=(ans+C(n+1,k)*pw[m+1-k][n]%P)%P;
				else ans=(ans+(P-C(n+1,k))*pw[m+1-k][n]%P)%P;
			}
			IO<<ans*invfac[n+1]%P<<'\n';
		}
	}
}//Metatron Mal'akh 绝灭天使·天翼

Subtask 5

因为 sub3 和 sub4 有点毁天灭地，所以先来讲 Sub5 了。

首先出题人显然把这玩意的 \(k+1\) 次幂强行拆开了，现在我们把它合起来，顺便把 \(k+1\) 转化成 \(k\)。

\[\begin{aligned} Ans&=\sum_{k=0}^{m}\sum_{i=0}^{m-k}\frac{\dbinom{2n+1}{i}(-1)^{m-k}(k+1)^{m+n+1-i}}{(m-k-i)!(k+1)!}\\ &=\sum_{k=1}^{m+1}\sum_{i=0}^{m-k+1}\frac{\dbinom{2n+1}{i}(-1)^{m-k}k^{m+n+1-i}}{(m-k+1-i)!k!}\\ \end{aligned} \]

继续考虑怎么化出斯特林数。

首先观测到 \(k=0\) 时，后面的式子也是 \(0\)​，所以可以把 \(k=0\) 扔进来。

天然的 \(\dfrac{k^{m+n+1-i}}{(m-k+1-i)!k!}\)​​​​ 肯定不要动，那么 \(i\)​​​​​​ 是固定的，考虑其余怎么配。

代入斯特林数的通项，我们可以发现：

\[\begin{Bmatrix}m+n+1-i\\m+1-i\end{Bmatrix}=\sum_k \frac{k^{m+n+1-i}(-1)^{m+1-k-i}}{k!(m-k+1-i)!} \]

直接拆 \((-1)^{m-k}\) 为 \((-1)^{m+1-k-i}(-1)^{i-1}\) 就可以了。

于是原式化为

\[\begin{aligned} Ans&=\sum_{k=0}^{m+1}\sum_{i=0}^{m-k+1}\frac{\dbinom{2n+1}{i}(-1)^{m-k}k^{m+n+1-i}}{(m-k+1-i)!k!}\\ &=\sum_{i=0}^{m}\sum_{k=0}^{m-i+1}\frac{\dbinom{2n+1}{i}(-1)^{m-k}k^{m+n+1-i}}{(m-k+1-i)!k!}\\ &=\sum_{i=0}^{m}\dbinom{2n+1}{i}(-1)^{i-1}\sum_{k=0}^{m-i+1}\frac{(-1)^{m+1-k-i}k^{m+n+1-i}}{(m-k+1-i)!k!}\\ &=\sum_{i=0}^{m}\dbinom{2n+1}{i}(-1)^{i-1}\begin{Bmatrix}n+k+1-i\\k+1-i\end{Bmatrix} \end{aligned} \]

注意到这正好是二阶欧拉数的通项，而注意到数据范围只有 \(2000\)，所以可以通过递归式 \(n^2\) dp 求得。

复杂度为 \(O(n^2)-O(1)\) 的。

namespace Satan{
	ll Euler[2005][2005];
	void Devil(ll n){
		rep(i,0,n)Euler[i][0]=1;
		rep(i,1,n)rep(j,1,i)Euler[i][j]=((j+1)*Euler[i-1][j]%P+(2*i-1-j)*Euler[i-1][j-1])%P;
	}
	void Satan(){
		for(IO>>q;q--;){
			IO>>n>>m;
			IO<<Euler[n][m]<<'\n';
		}
	}
}//Satan 反转体 救世魔王

Subtask 3

相比 Subtask 2 而言，数据范围上升到了 \(998244352\)​​，考虑怎么优化预处理。

快速阶乘算法我不会，所以上分块打表。

这里块长直接设 \(10^6\)​，这样数组长度只有 \(1000\)​。

为了防止快速幂被卡，所以上光速幂，块长设 \(32768\)​ 就可以了。

然后这个 sub 就没了。

const ll prefac[]={喵喵喵喵喵喵喵喵喵};
ll calcfac(ll n){
	ll ret=prefac[n/B];
	rep(i,n/B*B+1,n)ret=ret*i%P;
	return ret;
}
namespace Kadour{
	const ll bl=32768;
	ll pw1[25][32800],inv[25],pw2[25][32800];
	void Metatron(ll m){
		rep(i,1,m){
			pw1[i][0]=1;
			rep(j,1,bl)pw1[i][j]=pw1[i][j-1]*i%P;
			pw2[i][0]=1;
			rep(j,1,bl)pw2[i][j]=pw2[i][j-1]*pw1[i][bl]%P;
		}
		inv[0]=inv[1]=1;
		rep(i,2,m)inv[i]=(P-P/i)*inv[P%i]%P;
	}
	ll pw(ll n,ll m){
		return pw2[n][m>>15]*pw1[n][m&(bl-1)]%P;
	}
	void Kadour(){
		for(IO>>q;q--;){
			IO>>n>>m;
			ll ans=0,binom=1;
			rep(k,0,m){
				if(k&1^1)ans=(ans+binom*pw(m+1-k,n)%P)%P;
				else ans=(ans-binom*pw(m+1-k,n)%P+P)%P;
				binom=binom*(n+1-k)%P*inv[k+1]%P;
			}
			IO<<ans*qpow(calcfac(n+1),P-2)%P<<'\n';
		}
	}
}//Metatron Kadour 绝灭天使·光剑

Subtask 4

\[f_n=\sum_{k=0}^{n}k^n\frac{k^m}{k!}\frac{(-1)^{n-k}}{(n-k)!} \]

小 Q 对小 K 说：你这个菜鸡，这不显然的第二类斯特林数吗，你斯特林数怎么学的了。

然后他仔细看了一遍，

\[n\le 10^{10^5},m\le 10 \]

傻眼了，发现他不会这道题。

容易知道这道题要求的就是

\[\begin{Bmatrix}n+m\\n\end{Bmatrix} \]

然后根据具体数学：

\[\begin{Bmatrix}n+m\\n\end{Bmatrix}=\sum_{k=0}^{m}\left\langle\left\langle\begin{matrix}m\\k\end{matrix}\right\rangle\right\rangle \binom{n+2m-k}{2m} \]

那么 \(O(m^2)\)​ 预处理二阶欧拉数，然后用类似 Subtask 3 的做法处理组合数，每次变化都是 \(O(1)\) 的。

但是这里要求的逆元都很大，怎么办呢？考虑进行数据分治，对 \(31\sim 40\) 的数据预处理逆元，对 \(41\sim 50\) 的数据 \(O(\log P)\)​​ 计算。这样 \(31\sim 40\) 的复杂度是 \(O(m^2+n)-O(m)\) 的，而 \(41\sim 50\) 的复杂度是 \(O(m^2)-O(m\log P)\) 的，均可通过。

namespace Artelif{
	ll inv[N<<1],Euler[15][15],ma;
	void Metatron(ll n,ll m){
		ma=n;
		inv[0]=inv[1]=1;
		rep(i,2,n)inv[i]=(P-P/i)*inv[P%i]%P;
		rep(i,0,m)Euler[i][0]=1;
		rep(i,1,m)rep(j,1,i)Euler[i][j]=((j+1)*Euler[i-1][j]%P+(2*i-1-j)*Euler[i-1][j-1])%P;
	}
	ll calcinv(ll n){
		return n<=ma?inv[n]:qpow(n,P-2);
	}
	void Artelif(){
		for(IO>>q;q--;){
			IO.read(n,P);
			IO>>m;
			ll binom=1,ans=0;
			rep(i,n,n+2*m-1)binom=binom*i%P*calcinv(i-n+1)%P;
			rep(i,0,m){
				ans=(ans+binom*Euler[m][i]%P)%P;
				binom=binom*calcinv(n+2*m-1-i)%P*(n-i-1)%P;
			}
			IO<<ans<<'\n';
		}
	}
}//Metatron Artelif 绝灭天使·炮冠

下面是完整代码。

// Problem: P6073 『MdOI R1』Epic Convolution
// Contest: Luogu
// URL: https://www.luogu.com.cn/problem/P6073
// Memory Limit: 500 MB
// Time Limit: 400 ms
// 
// Powered by CP Editor (https://cpeditor.org)

#include<bits/stdc++.h>
#define endl '\n' 
#define rep(i,a,b) for(ll i=(a);i<=(b);++i)
#define Rep(i,a,b) for(ll i=(a);i>=(b);--i)
using namespace std;
typedef long long ll;
inline void chkmax(ll &x,ll y){if(x<y)x=y;}
inline void chkmin(ll &x,ll y){if(x>y)x=y;}
struct IO_Tp {
    const static int _I_Buffer_Size = 2 << 22;
    char _I_Buffer[_I_Buffer_Size], *_I_pos = _I_Buffer;

    const static int _O_Buffer_Size = 2 << 22;
    char _O_Buffer[_O_Buffer_Size], *_O_pos = _O_Buffer;

    IO_Tp() { fread(_I_Buffer, 1, _I_Buffer_Size, stdin); }
    ~IO_Tp() { fwrite(_O_Buffer, 1, _O_pos - _O_Buffer, stdout); }

	void read(ll &res,ll mod){
		ll f=1;
        while (!isdigit(*_I_pos)&&(*_I_pos)!='-') ++_I_pos;
        if(*_I_pos=='-')f=-1,++_I_pos;
        res = *_I_pos++ - '0';
        while (isdigit(*_I_pos)) res = (res * 10 + (*_I_pos++ - '0') ) % mod;
        res*=f;
	}

    IO_Tp &operator>>(ll &res) {
    	ll f=1;
        while (!isdigit(*_I_pos)&&(*_I_pos)!='-') ++_I_pos;
        if(*_I_pos=='-')f=-1,++_I_pos;
        res = *_I_pos++ - '0';
        while (isdigit(*_I_pos)) res = res * 10 + (*_I_pos++ - '0');
        res*=f;
        return *this;
    }

    IO_Tp &operator<<(ll n) {
    	if(n<0)*_O_pos++='-',n=-n;
        static char _buf[10];
        char *_pos = _buf;
        do
            *_pos++ = '0' + n % 10;
        while (n /= 10);
        while (_pos != _buf) *_O_pos++ = *--_pos;
        return *this;
    }

    IO_Tp &operator<<(char ch) {
        *_O_pos++ = ch;
        return *this;
    }
} IO;
typedef vector<int> vec;
const int N=2e5+5,P=998244353,B=1e6;
ll op,q,n,m;
const ll prefac[]={1,373341033,45596018,834980587,623627864,428937595,442819817,499710224,833655840,83857087,295201906,788488293,671639287,849315549,597398273,813259672,732727656,244038325,122642896,310517972,160030060,483239722,683879839,712910418,384710263,433880730,844360005,513089677,101492974,959253371,957629942,678615452,34035221,56734233,524027922,31729117,102311167,330331487,8332991,832392662,545208507,594075875,318497156,859275605,300738984,767818091,864118508,878131539,316588744,812496962,213689172,584871249,980836133,54096741,417876813,363266670,335481797,730839588,393495668,435793297,760025067,811438469,720976283,650770098,586537547,117371703,566486504,749562308,708205284,932912293,939830261,983699513,206579820,301188781,593164676,770845925,247687458,41047791,266419267,937835947,506268060,6177705,936268003,166873118,443834893,328979964,470135404,954410105,117565665,832761782,39806322,478922755,394880724,821825588,468705875,512554988,232240472,876497899,356048018,895187265,808258749,575505950,68190615,939065335,552199946,694814243,385460530,529769387,640377761,916128300,440133909,362216114,826373774,502324157,457648395,385510728,904737188,78988746,454565719,623828097,686156489,713476044,63602402,570334625,681055904,222059821,477211096,343363294,833792655,461853093,741797144,74731896,930484262,268372735,941222802,677432735,474842829,700451655,400176109,697644778,390377694,790010794,360642718,505712943,946647976,339045014,715797300,251680896,70091750,40517433,12629586,850635539,110877109,571935891,695965747,634938288,69072133,155093216,749696762,963086402,544711799,724471925,334646013,574791029,722417626,377929821,743946412,988034679,405207112,18063742,104121967,638607426,607304611,751377777,35834555,313632531,18058363,656121134,40763559,562910912,495867250,48767038,210864657,659137294,715390025,865854329,324322857,388911184,286059202,636456178,421290700,832276048,726437551,526417714,252522639,386147469,674313019,274769381,226519400,272047186,117153405,712896591,486826649,119444874,338909703,18536028,41814114,245606459,140617938,250512392,57084755,157807456,261113192,40258068,194807105,325341339,884328111,896332013,880836012,737358206,202713771,785454372,399586250,485457499,640827004,546969497,749602473,159788463,159111724,218592929,675932866,314795475,811539323,246883213,696818315,759880589,4302336,353070689,477909706,559289160,79781699,878094972,840903973,367416824,973366814,848259019,462421750,667227759,897917455,81800722,956276337,942686845,420541799,417005912,272641764,941778993,217214373,192220616,267901132,50530621,652678397,354880856,164289049,781023184,105376215,315094878,607856504,733905911,457743498,992735713,35212756,231822660,276036750,734558079,424180850,433186147,308380947,18333316,12935086,351491725,655645460,535812389,521902115,67016984,48682076,64748124,489360447,361275315,786336279,805161272,468129309,645091350,887284732,913004502,358814684,281295633,328970139,395955130,164840186,820902807,761699708,246274415,592331769,913846362,866682684,600130702,903837674,529462989,90612675,526540127,533047427,110008879,674279751,801920753,645226926,676886948,752481486,474034007,457790341,166813684,287671032,188118664,244731384,404032157,269766986,423996017,182948540,356801634,737863144,652014069,206068022,504569410,919894484,593398649,963768176,882517476,702523597,949028249,128957299,171997372,50865043,20937461,690959202,581356488,369182214,993580422,193500140,540665426,365786018,743731625,144980423,979536721,773259009,617053935,247670131,843705280,30419459,985463402,261585206,237885042,111276893,488166208,137660292,720784236,244467770,26368504,792857103,666885724,670313309,905683034,259415897,512017253,826265493,111960112,633652060,918048438,516432938,386972415,996212724,610073831,444094191,72480267,665038087,11584804,301029012,723617861,113763819,778259899,937766095,535448641,593907889,783573565,673298635,599533244,655712590,173350007,868198597,169013813,585161712,697502214,573994984,285943986,675831407,3134056,965907646,401920943,665949756,236277883,612745912,813282113,892454686,901222267,624900982,927122298,686321335,84924870,927606072,506664166,353631992,165913238,566073550,816674343,864877926,171259407,908752311,874007723,803597299,613676466,880336545,282280109,128761001,58852065,474075900,434816091,364856903,149123648,388854780,314693916,423183826,419733481,888483202,238933227,336564048,757103493,100189123,855479832,51370348,403061033,496971759,831753030,251718753,272779384,683379259,488844621,881783783,659478190,445719559,740782647,546525906,985524427,548033568,333772553,331916427,752533273,730387628,93829695,655989476,930661318,334885743,466041862,428105027,888238707,232218076,769865249,730641039,616996159,231721356,326973501,426068899,722403656,742756734,663270261,364187931,350431704,671823672,633125919,226166717,386814657,237594135,451479365,546182474,119366536,465211069,605313606,728508871,249619035,663053607,900453742,48293872,229958401,62402409,69570431,71921532,960467929,537087913,514588945,513856225,415497414,286592050,645469437,102052166,163298189,873938719,617583886,986843080,962390239,580971332,665147020,88900164,89866970,826426395,616059995,443012312,659160562,229855967,687413213,59809521,398599610,325666688,154765991,159186619,210830877,386454418,84493735,974220646,820097297,2191828,481459931,729073424,551556379,926316039,151357011,808637654,218058015,786112034,850407126,84202800,94214098,30019651,121701603,176055335,865461951,553631971,286620803,984061713,888573766,302767023,977070668,110954576,83922475,51568171,60949367,19533020,510592752,615419476,341370469,912573425,286207526,206707897,384156962,414163604,193301813,749570167,366933789,11470970,600191572,391667731,328736286,30645366,215162519,604947226,236199953,718439098,411423177,803407599,632441623,766760224,263006576,757681534,61082578,681666415,947466395,12206799,659767098,933746852,978860867,59215985,161179205,439197472,259779111,511621808,145770512,882749888,943124465,872053396,631078482,166861622,743415395,772287179,602427948,924112080,385643091,794973480,883782693,869723371,805963889,313106351,262132854,400034567,488248149,265769800,791715397,408753255,468381897,415812467,172922144,64404368,281500398,512318142,288791777,955559118,242484726,536413695,205340854,707803527,576699812,218525078,875554190,46283078,833841915,763148293,807722138,788080170,556901372,150896699,253151120,97856807,918256774,771557187,582547026,472709375,911615063,743371401,641382840,446540967,184639537,157247760,775930891,939702814,499082462,19536133,548753627,593243221,563850263,185475971,687419227,396799323,657976136,864535682,433009242,860830935,33107339,517661450,467651311,812398757,202133852,431839017,709549400,99643620,773282878,290471030,61134552,129206504,929147251,837008968,422332597,353775281,469563025,62265336,835064501,851685235,21197005,264793769,326416680,118842991,84257200,763248924,687559609,150907932,401832452,242726978,766752066,959173604,390269102,992293822,744816299,476631694,177284763,702429415,374065901,169855231,629007616,719169602,564737074,475119050,714502830,40993711,820235888,749063595,239329111,612759169,18591377,419142436,442202439,941600951,158013406,637073231,471564060,447222237,701248503,599797734,577221870,69656699,51052704,6544303,10958310,554955500,943192237,192526269,897983911,961628039,240232720,627280533,710239542,70255649,261743865,228474833,776408079,304180483,63607040,953297493,758058902,395529997,156010331,825833840,539880795,234683685,52626619,751843490,116909119,62806842,574857555,353417551,40061330,822203768,681051568,490913702,9322961,766631257,124794668,37844313,163524507,729108319,490867505,47035168,682765157,53842115,817965276,757179922,339238384,909741023,150530547,158444563,140949492,993302799,551621442,137578883,475122706,443869843,605400098,689361523,769596520,801661499,474900284,586624857,349960501,134084537,650564083,877097974,379857427,887890124,159436401,133274277,986182139,729720334,568925901,459461496,499309445,493171177,460958750,380694152,168836226,840160881,141116880,225064950,109618190,842341383,85305729,759273275,97369807,669317759,766247510,829017039,550323884,261274540,918239352,29606025,870793828,293683814,378510746,367270918,481292028,813097823,798448487,230791733,899305835,504040630,162510533,479367951,275282274,806951470,462774647,56473153,184659008,905122161,664034750,109726629,59372704,325795100,486860143,843736533,924723613,880348000,801252478,616515290,776142608,284803450,583439582,274826676,6018349,377403437,244041569,527081707,544763288,708818585,354033051,904309832,589922898,673933870,682858433,945260111,899893421,515264973,911685911,9527148,239480646,524126897,48259065,578214879,118677219,786127243,869205770,923276513,937928886,802186160,12198440,638784295,34200904,758925811,185027790,80918046,120604699,610456697,573601211,208296321,49743354,653691911,490750754,674335312,887877110,875880304,308360096,414636410,886100267,8525751,636257427,558338775,500159951,696213291,97268896,364983542,937928436,641582714,586211304,345265657,994704486,443549763,207259440,302122082,166055224,623250998,239642551,476337075,283167364,211328914,68064804,950202136,187552679,18938709,646784245,598764068,538505481,610424991,864445053,390248689,278395191,686098470,935957187,868529577,329970687,804930040,84992079,474569269,810762228,573258936,756464212,155080225,286966169,283614605,19283401,24257676,871831819,612689791,846988741,617120754,971716517,979541482,297910784,991087897,783825907,214821357,689498189,405026419,946731704,609346370,707669156,457703127,957341187,980735523,649367684,791011898,82098966,234729712,105002711,130614285,291032164,193188049,363211260,58108651,100756444,954947696,346032213,863300806,36876722,622610957,289232396,667938985,734886266,395881057,417188702,183092975,887586469,83334648,797819763,100176902,781587414,841864935,371674670,18247584};
ll qpow(ll a,ll b){return !b?1:qpow(a*a%P,b>>1)*(b&1?a:1)%P;}
////////////////////////Subtask 1///////////////////
namespace Shemesh{
	ll ans[N];
    void Metatron(){
        IO>>n>>q;
        rep(i,1,n-1)ans[i]=(ans[i-1]+qpow(i,n))%P;
    }
	void Shemesh(){
		while(q--){
			ll m;
			IO>>m;
			IO<<ans[m-1]<<'\n';
		}
	}
}// Metatron Shemesh 绝灭天使·日轮
////////////////////////Subtask 2///////////////////
namespace Malakh{
	ll fac[N],invfac[N],pw[25][N];
	void Metatron(ll n,ll m){
		fac[0]=1;
		rep(i,1,n)fac[i]=fac[i-1]*i%P;
		invfac[n]=qpow(fac[n],P-2);
		Rep(i,n-1,0)invfac[i]=invfac[i+1]*(i+1)%P;
		rep(i,1,m){
			pw[i][0]=1;
			rep(j,1,n)pw[i][j]=pw[i][j-1]*i%P;
		}
	}
	ll C(ll n,ll m){return fac[n]*invfac[m]%P*invfac[n-m]%P;}
	void Malakh(){
		for(IO>>q;q--;){
			IO>>n>>m;
			ll ans=0;
			rep(k,0,m){
				if(k&1^1)ans=(ans+C(n+1,k)*pw[m+1-k][n]%P)%P;
				else ans=(ans+(P-C(n+1,k))*pw[m+1-k][n]%P)%P;
			}
			IO<<ans*invfac[n+1]%P<<'\n';
		}
	}
}//Metatron Mal'akh 绝灭天使·天翼
////////////////////////Subtask 3///////////////////
ll calcfac(ll n){
	ll ret=prefac[n/B];
	rep(i,n/B*B+1,n)ret=ret*i%P;
	return ret;
}
namespace Kadour{
	const ll bl=32768;
	ll pw1[25][32800],inv[25],pw2[25][32800];
	void Metatron(ll m){
		rep(i,1,m){
			pw1[i][0]=1;
			rep(j,1,bl)pw1[i][j]=pw1[i][j-1]*i%P;
			pw2[i][0]=1;
			rep(j,1,bl)pw2[i][j]=pw2[i][j-1]*pw1[i][bl]%P;
		}
		inv[0]=inv[1]=1;
		rep(i,2,m)inv[i]=(P-P/i)*inv[P%i]%P;
	}
	ll pw(ll n,ll m){
		return pw2[n][m>>15]*pw1[n][m&(bl-1)]%P;
	}
	void Kadour(){
		for(IO>>q;q--;){
			IO>>n>>m;
			ll ans=0,binom=1;
			rep(k,0,m){
				if(k&1^1)ans=(ans+binom*pw(m+1-k,n)%P)%P;
				else ans=(ans-binom*pw(m+1-k,n)%P+P)%P;
				binom=binom*(n+1-k)%P*inv[k+1]%P;
			}
			IO<<ans*qpow(calcfac(n+1),P-2)%P<<'\n';
		}
	}
}//Metatron Kadour 绝灭天使·光剑
namespace Artelif{
	ll inv[N<<1],Euler[15][15],ma;
	void Metatron(ll n,ll m){
		ma=n;
		inv[0]=inv[1]=1;
		rep(i,2,n)inv[i]=(P-P/i)*inv[P%i]%P;
		rep(i,0,m)Euler[i][0]=1;
		rep(i,1,m)rep(j,1,i)Euler[i][j]=((j+1)*Euler[i-1][j]%P+(2*i-1-j)*Euler[i-1][j-1])%P;
	}
	ll calcinv(ll n){
		return n<=ma?inv[n]:qpow(n,P-2);
	}
	void Artelif(){
		for(IO>>q;q--;){
			IO.read(n,P);
			IO>>m;
			ll binom=1,ans=0;
			rep(i,n,n+2*m-1)binom=binom*i%P*calcinv(i-n+1)%P;
			rep(i,0,m){
				ans=(ans+binom*Euler[m][i]%P)%P;
				binom=binom*calcinv(n+2*m-1-i)%P*(n-i-1)%P;
			}
			IO<<ans<<'\n';
		}
	}
}//Metatron Artelif 绝灭天使·炮冠
////////////////////////Subtask 5///////////////////
namespace Satan{
	ll Euler[2005][2005];
	void Devil(ll n){
		rep(i,0,n)Euler[i][0]=1;
		rep(i,1,n)rep(j,1,i)Euler[i][j]=((j+1)*Euler[i-1][j]%P+(2*i-1-j)*Euler[i-1][j-1])%P;
	}
	void Satan(){
		for(IO>>q;q--;){
			IO>>n>>m;
			IO<<Euler[n][m]<<'\n';
		}
	}
}//Satan 反转体 救世魔王
int main(){
	IO>>op;
	if(op==1)Shemesh::Metatron(),Shemesh::Shemesh();
	else if(op==2)Malakh::Metatron(2e5+1,21),Malakh::Malakh();
	else if(op==3)Kadour::Metatron(21),Kadour::Kadour();
	else if(op==4)Artelif::Metatron(2e5+100,10),Artelif::Artelif();
	else if(op==5)Satan::Devil(2000),Satan::Satan();
	return 0;
}

完结撒折纸！