(mathcal{Description})
Link.
给定 (n,s,a_0,a_1,a_2,a_3)，求：
[sum_{i=0}^ninom{n}is^ia_{imod4}mod998244353
]
多测，数据组数 (le10^5)，(nle10^{18})，其余输入 (le10^8)。
(mathcal{Solution})
单位根反演板题。记一个函数 (f) 有：
[egin{aligned}
f(x)&=sum_{i=0}^ninom{n}is^ix^i\
&=(sx+1)^n
end{aligned}
]
问题即求 (imod4=0,1,2,3) 时 (a_i) 倍 ([x^i]f(x)) 之和。以 (imod4=0) 为例：
[egin{aligned}
sum_{i=0}^n[4|i]a_0[x^i]f(x)&=frac{1}4a_0sum_{i=0}^nleft(sum_{j=0}^3omega_4^{ij}
ight)inom{n}is^i\
&=frac{1}4a_0sum_{j=0}^3f(omega_4^j)
end{aligned}
]
直接代四个单位根进去算出来即可。对于其他三个 (imod4) 的值，将 (f) 的各系数位移就能类似地求出答案。
复杂度 (mathcal O(Tlog n))（( imes4^2) 的常数）。
(mathcal{Code})
/* Clearink */
#include <cstdio>
#define rep( i, l, r ) for ( int i = l, repEnd##i = r; i <= repEnd##i; ++i )
#define per( i, r, l ) for ( int i = r, repEnd##i = l; i >= repEnd##i; --i )
typedef long long LL;
inline LL rint () {
LL x = 0, f = 1; char s = getchar ();
for ( ; s < '0' || '9' < s; s = getchar () ) f = s == '-' ? -f : f;
for ( ; '0' <= s && s <= '9'; s = getchar () ) x = x * 10 + ( s ^ '0' );
return x * f;
}
template<typename Tp>
inline void wint ( Tp x ) {
if ( x < 0 ) putchar ( '-' ), x = -x;
if ( 9 < x ) wint ( x / 10 );
putchar ( x % 10 ^ '0' );
}
const int MOD = 998244353, G = 3, INV4 = 748683265;
LL n;
int w[4], s, a[4];
inline int mul ( const long long a, const int b ) { return a * b % MOD; }
inline int add ( int a, const int b ) { return ( a += b ) < MOD ? a : a - MOD; }
inline int mpow ( int a, int b ) {
int ret = 1;
for ( ; b; a = mul ( a, a ), b >>= 1 ) ret = mul ( ret, b & 1 ? a : 1 );
return ret;
}
inline int f ( const int x ) {
return mpow ( add ( mul ( s, x ), 1 ), n );
}
int main () {
w[0] = 1, w[1] = mpow ( G, MOD - 1 >> 2 );
w[2] = mul ( w[1], w[1] ), w[3] = mul ( w[2], w[1] );
for ( int T = rint (); T--; ) {
n = rint () % ( MOD - 1 ), s = rint ();
rep ( i, 0, 3 ) a[i] = rint ();
int ans = 0;
rep ( r, 0, 3 ) {
int res = 0;
rep ( i, 0, 3 ) {
res = add ( res,
mul ( f ( w[i] ), mpow ( w[r * i & 3], MOD - 2 ) ) );
}
ans = add ( ans, mul ( res, a[r] ) );
}
wint ( mul ( ans, INV4 ) ), putchar ( '
' );
}
return 0;
}