#6485. LJJ 学二项式定理
$$
\begin {aligned}
& \sum _ {i = 0} ^ n \binom n i s _ i a _ {i \bmod 4} \\
= & \sum _ {i = 0} ^ n \binom n i s _ i \sum _ {j = 0} ^ 3 a _ j [4 | i - j] \\
= & \sum _ {i = 0} ^ n \binom n i s _ i \sum _ {j = 0} ^ 3 a _ j \frac {\sum _ {k = 0} ^ 3 \omega _ 4 ^ {k(i - j)}} 4 \\
= & \frac {\sum _ {i = 0} ^ n \binom n i s _ i \sum _ {j = 0} ^ 3 a _ j \sum _ {k = 0} ^ 3 \omega _ 4 ^ {k(i - j)}} 4 \\
= & \frac {\sum _ {j = 0} ^ 3 a _ j \sum _ {k = 0} ^ 3 \omega _ 4 ^ {-kj} \sum _ {i = 0} ^ n \binom n i ({s \omega _ 4 ^ k}) ^ i } 4\\
= & \frac {\sum _ {j = 0} ^ 3 a _ j \sum _ {k = 0} ^ 3 \omega _ 4 ^ {-kj} ({s \omega _ 4 ^ k + 1} ^ n)} 4\\
\end {aligned}
$$
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| #include <cstdio>
using namespace std;
template <class Type>
void read(Type &x)
{
char c;
bool flag = false;
while ((c = getchar()) < '0' || c > '9')
c == '-' && (flag = true);
x = c - '0';
while ((c = getchar()) >= '0' && c <= '9')
x = (x << 3) + (x << 1) + c - '0';
if (flag) x = ~x + 1;
}
template <class Type, class ...rest>
void read(Type &x, rest &...y) { read(x), read(y...); }
template <class Type>
void write(Type x)
{
if (x < 0) putchar('-'), x = ~x + 1;
if (x > 9) write(x / 10);
putchar(x % 10 + '0');
}
typedef long long LL;
const int mod = 998244353;
void adj (int &x) { x += x >> 31 & mod; }
int binpow (int b, LL k = mod - 2)
{
int res = 1;
for (; k; k >>= 1, b = (LL)b * b % mod)
if (k & 1) res = (LL)res * b % mod;
return res;
}
struct ModInt
{
int x;
ModInt (int _ = 0) { adj(x = _); }
int operator () () const { return x; }
ModInt& operator += (const ModInt &_) { adj(x += _.x - mod); return *this; }
ModInt& operator -= (const ModInt &_) { adj(x -= _.x); return *this; }
ModInt& operator *= (const ModInt &_) { x = (LL)x * _.x % mod; return *this; }
ModInt& operator /= (const ModInt &_) { x = (LL)x * binpow(_.x) % mod; return *this; }
ModInt& operator ^= (const LL &_) { x = binpow(x, _); return *this; }
ModInt operator + (const ModInt &_) const { ModInt res = x; res += _; return res; }
ModInt operator - (const ModInt &_) const { ModInt res = x; res -= _; return res; }
ModInt operator * (const ModInt &_) const { ModInt res = x; res *= _; return res; }
ModInt operator / (const ModInt &_) const { ModInt res = x; res /= _; return res; }
ModInt operator ^ (const LL &_) const { ModInt res = x; res ^= _; return res; }
};
int main ()
{
ModInt w(binpow(3, (mod - 1) / 4));
int T;
read(T);
for (LL n, s; T; --T)
{
read(n, s);
ModInt res;
for (int i = 0, a; i < 4; ++i)
{
read(a);
ModInt t;
for (int j = 0; j < 4; ++j)
t += (((w ^ j) * s + 1) ^ n) / (w ^ (i * j));
res += t * a;
}
write((res / 4)()), puts("");
}
return 0;
}
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