51Nod 1228 Bernoulli number

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51Nod 1228 Bernoulli number

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Test instructions

S (k,n) =1^k+2^k+...+n^k
The power of natural numbers and the modulus of 1e9+7 are obtained.
Limit:
1<= n <= 10^18; 1 <= k <= 2000
Ideas:
Bernoulli number
S (k,n) =s (k,n) =1/(k+1) * (c (k+1,k) *b[k]* (n+1) ^1 + C (k+1,k-1) *b[k-1]* (n+1) ^2 + ... + c (k+1,0) *b[0]* (n+1) ^ (k+1)) (B[i] is the Bernoulli number)
and B[n] are:
B[N]=-1/(n+1) * (c (n+1,0) *b[0] + C (n+1,1) *b[1] + ... + c (n+1,n-1) *b[n-1])

So b[0] ... B[k] can be preprocessed by O (k^2), and then the O (k) can be calculated for each s (k,n).


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/*51nod 1228
Test instructions
S (k,n) =1^k+2^k+...+n^k
The power of natural numbers and the modulus of 1e9+7 are obtained.
Limit:
1<= n <= 10^18; 1 <= k <= 2000
Ideas:
Bernoulli number
S (k,n) =s (k,n) =1/(k+1) * (c (k+1,k) *b[k]* (n+1) ^1 + C (k+1,k-1) *b[k-1]* (n+1) ^2 + ... + c (k+1,0) *b[0]* (n+1) ^ (k+1)) (B[i] for last Number of profits)
and B[n] are:
B[N]=-1/(n+1) * (c (n+1,0) *b[0] + C (n+1,1) *b[1] + ... + c (n+1,n-1) *b[n-1])
So b[0] ... B[k] can be preprocessed by O (k^2), and then the O (k) can be calculated for each s (k,n).
*/
#include<iostream>
#include<cstdio>
usingnamespaceStd
#defineLL__int64
ConstintMOD =1000000007;
ll EXT_GCD (ll A, ll B, LL &x, LL &y)
{
if(b = =0)
{
x =1, y =0;
returnA
}
LL ret = EXT_GCD (b, a% B, y, x);
Y-= A/b * x;
returnRet
}

ll Inv (ll A,intMSeeking inverse element
{
ll d, x, y, t = (LL) m;
D = EXT_GCD (A, t, X, y);
if(d = =1)return(x% t + t)% T;
return-1;
}

ConstintN =2005;

LL B[n], c[n][n];

voidInit ()
{
for(inti =0; i < N; ++i)
c[i][0] = C[i][i] =1;
for(inti =2; i < N; ++i)
for(intj =1; J < N; ++J)
C[I][J] = (C[i-1][J] + c[i-1][j-1])% MOD;
b[0] =1;
for(inti =1; i < N; ++i)
{
LL tmp =0;
for(intj =0; J < I; ++J)
TMP = (tmp + c[i +1][J] * b[j])% MOD;
B[i] = (TMP *-(INV (i +1, MoD)% mod + MoD)% MoD;
}
}

LL P[n];

voidGao (ll N, ll K)
{
p[0] =1;
for(inti =1; I <= K +1; ++i)
P[i] = (P[i-1] * ((n +1) (% MoD))% MoD;
LL ans =0;
for(inti =0; I <= K; ++i)
Ans = (ans + c[k +1][i] * B[i]% MOD * p[k +1-i])% MOD;
Ans = (ans * INV (k +1, MoD)% mod + MoD)% MoD;
printf"%i64d\n", ans);
}

intMain ()
{
Init ();
intT
LL N, K;
scanf"%d", &t);
while(t--)
{
scanf"%i64d%i64d", &n, &k);
Gao (n, K);
}
return0;
}

51Nod 1228 Bernoulli number

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