听说这个FFT跑得巨jb快

#pragma GCC target ("avx2")

#include <immintrin.h>

#include<bits/stdc++.h>

using namespace std;

typedef unsigned char U8;
typedef int I32;
typedef unsigned int U32;
typedef long long I64;
typedef unsigned long long U64;

const int P=998244353;
const int inv2=(P+1)>>1;
const int G=3;
const int GInv=332748118;
const int MaxExp=22;
const int MAXN=1048576<<1;
const int MAXK=500005;
const int BmM=288737297;
const int BmW=29;

struct FastIO{
    static const int S=1e7;
    int wpos;
    char wbuf[S];
    FastIO():wpos(0){}
    inline int xchar()
    {
        static char buf[S];
        static int len=0,pos=0;
        if(pos==len)
            pos=0,len=fread(buf,1,S,stdin);
        if(pos==len) exit(0);
        return buf[pos++];
    }
    inline int operator () ()
    {
        int c=xchar(),x=0;
        while (c<=32) c=xchar();
        for (;'0'<=c&&c<='9';c=xchar()) x=x*10+c-'0';
        return x;
    }
    inline I64 operator ! ()
    {
        int c=xchar();I64 x=0;
        while (c<=32) c=xchar();
        for (; '0'<=c&&c<='9';c=xchar()) x=x*10+c-'0';
        return x;
    }
    inline void wchar(int x)
    {
        if (wpos==S) fwrite(wbuf,1,S,stdout),wpos=0;
        wbuf[wpos++]=x;
    }
    inline void operator () (I64 x)
    {
        if (x < 0) wchar('-'),x=-x;
        char s[24];
        int n=0;
        while(x||!n) s[n++]='0'+x%10,x/=10;
        while(n--) wchar(s[n]);
        wchar('
');
    }
    inline void space(I64 x)
    {
        if (x<0) wchar('-'),x=-x;
        char s[24];
        int n=0;
        while(x||!n) s[n++]='0'+x%10,x/=10;
        while(n--) wchar(s[n]);
        wchar(' ');
    }
    inline void nextline()
    {
        wchar('
');
    }
    ~FastIO()
    {
        if (wpos) fwrite(wbuf, 1, wpos, stdout), wpos = 0;
    }
}io;

U32 GcdEx(U32 A, U32 B, I32& x, I32& y) {
    if (!B) {
        x = 1;
        y = 0;
        return A;
    }
    U32 d = GcdEx(B, A % B, y, x);
    y -= x * (I32) (A / B);
    return d;
}

inline U32 MNorm(I32 V) {
    V = V % P;
    return (U32) (V < 0 ? V + P : V);
}

inline U32 MAdd(U32 A, U32 B) {
    U32 res = A + B;
    return res < P ? res : res - P;
}

inline U32 MSub(U32 A, U32 B) {
    U32 res = A - B;
    return A < B ? res + P : res;
}

inline U32 MMul(U32 A, U32 B) {
    return (U32) ((U64) A * B % P);
}

inline U32 MPow(U32 A, U32 B) {
    U32 res = 1;
    while (B) {
        if (B & 1)
            res = MMul(res, A);
        A = MMul(A, A);
        B >>= 1;
    }
    return res;
}

inline U32 MInv(U32 N) {
    I32 x, y;
    GcdEx(N, P, x, y);
    x %= P;
    return (U32) (x < 0 ? x + P : x);
}

inline __m256i VLod(const U32* __restrict__ A) {
    return _mm256_load_si256((const __m256i*) A);
}

inline void VSto(U32* __restrict__ A, __m256i v) {
    _mm256_store_si256((__m256i*) A, v);
}

inline __m256i VEx0(__m256i v) {
    const __m256i vm0 = _mm256_set_epi64x(
        0x111111111b1a1918, 0x1111111113121110,
        0x111111110b0a0908, 0x1111111103020100
    );
    return _mm256_shuffle_epi8(v, vm0);
}

inline __m256i VEx1(__m256i v) {
    const __m256i vm1 = _mm256_set_epi64x(
        0x111111111f1e1d1c, 0x1111111117161514,
        0x111111110f0e0d0c, 0x1111111107060504
    );
    return _mm256_shuffle_epi8(v, vm1);
}

inline __m256i VIntlv(__m256i v0, __m256i v1) {
    return _mm256_blend_epi32(v0, _mm256_shuffle_epi32(v1, 0xb1), 0xaa);
}

inline __m256i VAdd(__m256i va, __m256i vb) {
    const __m256i vm32 = _mm256_set1_epi32(P);
    __m256i vra = _mm256_add_epi32(va, vb);
    __m256i vrb = _mm256_sub_epi32(vra, vm32);
    return _mm256_min_epu32(vra, vrb);
}

inline __m256i VSub(__m256i va, __m256i vb) {
    const __m256i vm32 = _mm256_set1_epi32(P);
    __m256i vra = _mm256_sub_epi32(va, vb);
    __m256i vrb = _mm256_add_epi32(vra, vm32);
    return _mm256_min_epu32(vra, vrb);
}

inline __m256i VMul(__m256i va0, __m256i va1, __m256i vb0, __m256i vb1) {
    const __m256i vm32 = _mm256_set1_epi32(P);
    const __m256i vm64 = _mm256_set1_epi64x(P);
    const __m256i vbmm = _mm256_set1_epi64x(BmM);
    __m256i vmul0 = _mm256_mul_epi32(va0, vb0);
    __m256i vmul1 = _mm256_mul_epi32(va1, vb1);
    __m256i vlow = VIntlv(vmul0, vmul1);
    __m256i vquo0 = _mm256_srli_epi64(_mm256_mul_epi32(_mm256_srli_epi64(vmul0, 29), vbmm), BmW);
    __m256i vquo1 = _mm256_srli_epi64(_mm256_mul_epi32(_mm256_srli_epi64(vmul1, 29), vbmm), BmW);
    __m256i vval0 = _mm256_mul_epi32(vquo0, vm64);
    __m256i vval1 = _mm256_mul_epi32(vquo1, vm64);
    __m256i vval = VIntlv(vval0, vval1);
    __m256i vra = _mm256_sub_epi32(vlow, vval);
    __m256i vrb = _mm256_add_epi32(vra, vm32);
    __m256i vrc = _mm256_sub_epi32(vra, vm32);
    __m256i vmin = _mm256_min_epu32(vra, vrb);
    return _mm256_min_epu32(vmin, vrc);
}

inline __m256i VMul(__m256i va, __m256i vb0, __m256i vb1) {
    return VMul(VEx0(va), VEx1(va), vb0, vb1);
}

inline __m256i VMul(__m256i va, __m256i vb) {
    return VMul(va, VEx0(vb), VEx1(vb));
}

inline void VMul(U32* __restrict__ A, U32 Len, U32 W) {
    if (Len < 8) {
        for (U32 i = 0; i < Len; ++i)
            A[i] = MMul(A[i], W);
        return;
    }
    __m256i vw = _mm256_set1_epi64x(W);
    for (U32 i = 0; i < Len; i += 8)
        VSto(A + i, VMul(VLod(A + i), vw, vw));
}

inline void VMul(U32* __restrict__ A, const U32* __restrict__ B, U32 Len) {
    if (Len < 8) {
        for (U32 i = 0; i < Len; ++i)
            A[i] = MMul(A[i], B[i]);
        return;
    }
    for (U32 i = 0; i < Len; i += 8)
        VSto(A + i, VMul(VLod(A + i), VLod(B + i)));
}

inline void VSqr(U32* __restrict__ A, U32 Len) {
    if (Len < 8) {
        for (U32 i = 0; i < Len; ++i)
            A[i] = MMul(A[i], A[i]);
        return;
    }
    for (U32 i = 0; i < Len; i += 8) {
        __m256i va = VLod(A + i);
        __m256i v0 = VEx0(va);
        __m256i v1 = VEx1(va);
        VSto(A + i, VMul(v0, v1, v0, v1));
    }
}

U32 WbFwd[MaxExp + 1];
U32 WbInv[MaxExp + 1];
U32 LenInv[MaxExp + 1];

inline void NttInitAll(int Max) {
    for (int Exp = 0; Exp <= Max; ++Exp) {
        WbFwd[Exp] = MPow(G, (P - 1) >> Exp);
        WbInv[Exp] = MPow(GInv, (P - 1) >> Exp);
        LenInv[Exp] = MInv(1u << Exp);
    }
}

inline void NttImpl1(U32* __restrict__ A, U32 Len) {
    for (U32 j = 0; j < Len; j += 2) {
        U32 a0 = MAdd(A[j + 0], A[j + 1]);
        U32 b0 = MSub(A[j + 0], A[j + 1]);
        A[j + 0] = a0;
        A[j + 1] = b0;
    }
}

inline void NttFwd2(U32* __restrict__ A, U32 Len, U32 Wn) {
    for (U32 j = 0; j < Len; j += 4) {
        U32 a0 = MAdd(A[j + 0], A[j + 2]);
        U32 a1 = MAdd(A[j + 1], A[j + 3]);
        U32 b0 = MSub(A[j + 0], A[j + 2]);
        U32 b1 = MSub(A[j + 1], A[j + 3]);
        A[j + 0] = a0;
        A[j + 1] = a1;
        A[j + 2] = b0;
        A[j + 3] = MMul(b1, Wn);
    }
}

inline void NttFwd3(U32* __restrict__ A, U32 Len, U32 Wn) {
    U32 W2 = MMul(Wn, Wn);
    U32 W3 = MMul(W2, Wn);
    const __m128i vm32 = _mm_set1_epi32(P);
    for (U32 j = 0; j < Len; j += 8) {
        __m128i va = _mm_load_si128((const __m128i*) (A + j));
        __m128i vb = _mm_load_si128((const __m128i*) (A + j + 4));
        __m128i vc = _mm_add_epi32(va, vb);
        __m128i vd = _mm_sub_epi32(va, vb);
        __m128i ve = _mm_sub_epi32(vc, _mm_andnot_si128(_mm_cmpgt_epi32(vm32, vc), vm32));
        __m128i vf = _mm_add_epi32(vd, _mm_and_si128(_mm_cmpgt_epi32(vb, va), vm32));
        _mm_store_si128((__m128i*) (A + j), ve);
        _mm_store_si128((__m128i*) (A + j + 4), vf);
        A[j + 5] = MMul(Wn, A[j + 5]);
        A[j + 6] = MMul(W2, A[j + 6]);
        A[j + 7] = MMul(W3, A[j + 7]);
    }
}

inline void NttFwd(U32* __restrict__ A, int Exp) {
    U32 Len = 1u << Exp;
    U32 Wn = WbFwd[Exp];
    for (int i = Exp - 1; i >= 3; --i) {
        U32 ChkSiz = 1u << i;
        U32 tw2 = MMul(Wn, Wn);
        U32 tw3 = MMul(tw2, Wn);
        U32 tw4 = MMul(tw3, Wn);
        U32 tw5 = MMul(tw4, Wn);
        U32 tw6 = MMul(tw5, Wn);
        U32 tw7 = MMul(tw6, Wn);
        U32 twn = MMul(tw7, Wn);
        __m256i vw32 = _mm256_set_epi32(tw7, tw6, tw5, tw4, tw3, tw2, Wn, 1);
        __m256i vwn = _mm256_set1_epi64x(twn);
        for (U32 j = 0; j < Len; j += 2u << i) {
            U32* A_ = A + j;
            U32* B_ = A_ + ChkSiz;
            __m256i vw = vw32;
            for (U32 k = 0; k < ChkSiz; k += 8) {
                __m256i va = VLod(A_ + k);
                __m256i vb = VLod(B_ + k);
                __m256i vw0 = VEx0(vw);
                __m256i vw1 = VEx1(vw);
                __m256i vc = VAdd(va, vb);
                __m256i vd = VSub(va, vb);
                VSto(A_ + k, vc);
                VSto(B_ + k, VMul(vd, vw0, vw1));
                vw = VMul(vw0, vw1, vwn, vwn);
            }
        }
        Wn = MMul(Wn, Wn);
    }
    if (Exp >= 3) {
        NttFwd3(A, Len, Wn);
        Wn = MMul(Wn, Wn);
    }
    if (Exp >= 2)
        NttFwd2(A, Len, Wn);
    if (Exp)
        NttImpl1(A, Len);
}

inline void NttInv2(U32* __restrict__ A, U32 Len, U32 Wn) {
    for (U32 j = 0; j < Len; j += 4) {
        U32 a0 = A[j + 0];
        U32 a1 = A[j + 1];
        U32 b0 = A[j + 2];
        U32 b1 = MMul(A[j + 3], Wn);
        A[j + 0] = MAdd(a0, b0);
        A[j + 1] = MAdd(a1, b1);
        A[j + 2] = MSub(a0, b0);
        A[j + 3] = MSub(a1, b1);
    }
}

inline void NttInv3(U32* __restrict__ A, U32 Len, U32 Wn) {
    U32 W2 = MMul(Wn, Wn);
    U32 W3 = MMul(W2, Wn);
    const __m128i vm32 = _mm_set1_epi32(P);
    for (U32 j = 0; j < Len; j += 8) {
        A[j + 5] = MMul(Wn, A[j + 5]);
        A[j + 6] = MMul(W2, A[j + 6]);
        A[j + 7] = MMul(W3, A[j + 7]);
        __m128i va = _mm_load_si128((const __m128i*) (A + j));
        __m128i vb = _mm_load_si128((const __m128i*) (A + j + 4));
        __m128i vc = _mm_add_epi32(va, vb);
        __m128i vd = _mm_sub_epi32(va, vb);
        __m128i ve = _mm_sub_epi32(vc, _mm_andnot_si128(_mm_cmpgt_epi32(vm32, vc), vm32));
        __m128i vf = _mm_add_epi32(vd, _mm_and_si128(_mm_cmpgt_epi32(vb, va),vm32));
        _mm_store_si128((__m128i*) (A + j), ve);
        _mm_store_si128((__m128i*) (A + j + 4), vf);
    }
}

inline void NttInv(U32* __restrict__ A, int Exp) {
    if (!Exp)
        return;
    U32 Len = 1u << Exp;
    NttImpl1(A, Len);
    if (Exp == 1) {
        VMul(A, Len, LenInv[1]);
        return;
    }
    U32 Ws[MaxExp];
    Ws[0] = WbInv[Exp];
    for (int i = 1; i < Exp; ++i)
        Ws[i] = MMul(Ws[i - 1], Ws[i - 1]);
    NttInv2(A, Len, Ws[Exp - 2]);
    if (Exp == 2) {
        VMul(A, Len, LenInv[2]);
        return;
    }
    NttInv3(A, Len, Ws[Exp - 3]);
    if (Exp == 3) {
        VMul(A, Len, LenInv[3]);
        return;
    }
    for (int i = 3; i < Exp; ++i) {
        U32 ChkSiz = 1u << i;
        U32 Wn = Ws[Exp - 1 - i];
        U32 tw2 = MMul(Wn, Wn);
        U32 tw3 = MMul(tw2, Wn);
        U32 tw4 = MMul(tw3, Wn);
        U32 tw5 = MMul(tw4, Wn);
        U32 tw6 = MMul(tw5, Wn);
        U32 tw7 = MMul(tw6, Wn);
        U32 twn = MMul(tw7, Wn);
        __m256i vw32 = _mm256_set_epi32(tw7, tw6, tw5, tw4, tw3, tw2, Wn, 1);
        __m256i vwn = _mm256_set1_epi64x(twn);
        for (U32 j = 0; j < Len; j += 2u << i) {
            U32* A_ = A + j;
            U32* B_ = A_ + ChkSiz;
            __m256i vw = vw32;
            for (U32 k = 0; k < ChkSiz; k += 8) {
                __m256i vw0 = VEx0(vw);
                __m256i vw1 = VEx1(vw);
                __m256i vb = VMul(VLod(B_ + k), vw0, vw1);
                vw = VMul(vw0, vw1, vwn, vwn);
                __m256i va = VLod(A_ + k);
                __m256i vc = VAdd(va, vb);
                __m256i vd = VSub(va, vb);
                VSto(A_ + k, vc);
                VSto(B_ + k, vd);
            }
        }
    }
    VMul(A, Len, LenInv[Exp]);
}

inline int Log2Ceil(U32 N) {
    static const U8 Table[32] = {
        0,  9,  1,  10, 13, 21, 2,  29,
        11, 14, 16, 18, 22, 25, 3,  30,
        8,  12, 20, 28, 15, 17, 24, 7,
        19, 27, 23, 6,  26, 5,  4,  31,
    };
    N = (N << 1) - 1;
    N |= N >> 1;
    N |= N >> 2;
    N |= N >> 4;
    N |= N >> 8;
    N |= N >> 16;
    return (int) Table[(N * 0x07c4acddu) >> 27];
}

U32 f[MAXN],h[MAXN];

void clear(U32 *a,int n)
{
    int tot=1;
    while(tot<n) tot<<=1;
    memset(a,0,sizeof(U32)*tot);
}

namespace poly_multiply{U32 t1[MAXN],t2[MAXN],t3[MAXN],t4[MAXN];}
namespace poly_inverse{U32 t1[MAXN],t2[MAXN];}
namespace poly_sqrt{U32 t1[MAXN],t2[MAXN];}

void Mul(U32 *a,U32 *b,U32 *Res,int n)
{
    using namespace poly_multiply;
    int tot=1,lg2=0;
    while(tot<2*n) tot<<=1,lg2++;
    clear(t1,tot);clear(t2,tot);clear(Res,tot);
    for(int i=0;i<n;i++) t1[i]=a[i],t2[i]=b[i];
    NttFwd(t1,lg2);NttFwd(t2,lg2);
    for(int i=0;i<tot;i++) Res[i]=MMul(t1[i],t2[i]);
    NttInv(Res,lg2);
}

void Mul(U32 *a,U32 *b,U32 *c,U32 *Res,int n)
{
    using namespace poly_multiply;
    int tot=1,lg2=0;
    while(tot<2*n) tot<<=1,lg2++;
    clear(t1,tot);clear(t2,tot);clear(t3,tot);clear(Res,tot);
    for(int i=0;i<n;i++) t1[i]=a[i],t2[i]=b[i],t3[i]=c[i];
    NttFwd(t1,lg2);NttFwd(t2,lg2);NttFwd(t3,lg2);
    for(int i=0;i<tot;i++) Res[i]=MMul(MMul(t1[i],t2[i]),t3[i]);
    NttInv(Res,lg2);
}

void Inv(U32 *a,U32 *Res,int n)
{
    using namespace poly_inverse;
    clear(Res,n);clear(t1,n);clear(t2,n);
    Res[0]=MPow(a[0],P-2);
    int l=1;
    while(l<n)
    {
        for(int i=0;i<l;i++) t1[i]=MAdd(Res[i],Res[i]);
        l<<=1;
        Mul(Res,Res,a,t2,l);
        for(int i=0;i<l;i++) Res[i]=MSub(t1[i],t2[i]);
    }
}

void Sqrt(U32 *a,U32 *Res,int n)
{
    using namespace poly_sqrt;
    clear(Res,n);clear(t1,n);clear(t2,n);
    int l=1,tot=1;Res[0]=1;//a[0]=1
    while(tot<=n) tot<<=1;
    while(l<tot)
    {
        l<<=1;
        Mul(Res,Res,t1,l);Inv(Res,t2,l);
        for(int i=0;i<l;i++) t1[i]=MMul(MAdd(t1[i],a[i]),inv2);
        Mul(t1,t2,Res,l);
    }
}

void init()
{
    NttInitAll(21);
    f[2]=1;f[1]=MSub(0,6);f[0]=1;
    int t=clock();
    Sqrt(f,h,MAXK);
    for(int i=0;i<=MAXK;i++) h[i]=MSub(0,h[i]);
    h[0]++;h[1]--;
    for(int i=0;i<=MAXK;i++) h[i]=MMul(h[i],inv2);
}

int main()
{
    freopen("grid.in","r",stdin);
    freopen("grid.out","w",stdout);
    int T,n;
    for(T=io(),init();T--;n=io(),io(h[n]));
}