数学题

也许更好的阅读体验

(mathcal{Description})

给(n,b,k),求
(egin{aligned}sum_{i=1}^negin{pmatrix}i\k end{pmatrix}b^iend{aligned})

答案对(998244353)取模

(1leq kleq 500000,1leq nleq 10^{18},2leq bleq 998244353)

(mathcal{Solution})

设
(egin{aligned}f_{k}=sum ^{n}_{i=1}egin{pmatrix} i \ k end{pmatrix}b^{i}end{aligned})
则有
(egin{aligned}f_{k-1}=sum ^{n}_{i=1}egin{pmatrix} i \ k-1 end{pmatrix}b^{i}end{aligned})

(egin{aligned}bf_{k}=sum ^{n+1}_{i=2}egin{pmatrix} i-1 \ k end{pmatrix}b^{i}end{aligned})

(egin{aligned}bf_{k-1}=sum ^{n+1}_{i=2}egin{pmatrix} i-1 \ k-1 end{pmatrix}b^{i}end{aligned})

(egin{aligned}bf_{k}+bf_{k-1}=sum ^{n+1}_{i=2}left(egin{pmatrix} i-1 \ k end{pmatrix}+egin{pmatrix} i-1 \ k-1 end{pmatrix} ight)b^{i}=sum ^{n+1}_{i=2}egin{pmatrix} i \ k end{pmatrix}b^{i}=f_k+egin{pmatrix}n+1\ kend{pmatrix}b^{n+1}-egin{pmatrix}1\kend{pmatrix}bend{aligned})

移项得

(left(b-1 ight)f_k=egin{pmatrix}n+1\kend{pmatrix}b^{n+1}-egin{pmatrix}1\kend{pmatrix}b-bf_{k-1})

(f_k=dfrac{egin{pmatrix}n+1\kend{pmatrix}b^{n+1}-egin{pmatrix}1\kend{pmatrix}b-bf_{k-1}}{b-1})

当(k=0)时，要求的就是一个等比数列，我们用等比数列求和公式直接算就可以了

(f_0=dfrac{b^{n+1}-b}{b-1})

总复杂度(O(k))

(mathcal{Code})

/*******************************
Author:Morning_Glory
LANG:C++
Created Time:2019年09月17日 星期二 10时54分50秒
*******************************/
#include <cstdio>
#include <fstream>
#define ll long long
using namespace std;
const int mod = 998244353;
const int maxn = 500005;
ll n,b,k,ans;
ll invb,mi,C;
ll inv[maxn];
//{{{ksm
ll ksm (ll a,ll b)
{
	a%=mod;
	ll s=1;
	for (;b;b>>=1,a=a*a%mod)
		if (b&1)	s=s*a%mod;
	return s;
}
//}}}
int main()
{
	scanf("%lld%lld%lld",&n,&b,&k);
	C=(n+1)%mod,inv[1]=1,invb=ksm(b-1,mod-2);
	for (int i=2;i<=k;++i)	inv[i]=(-mod/i*inv[mod%i]%mod+mod)%mod;
	ans=(ksm(b,n+1)-b+mod)%mod*invb%mod;
	mi=ksm(b,n+1);
	if (n<k){	printf("0
");return 0;}
	if (n==k){	printf("%lld
",ksm(b,n));return 0;}
	for (int i=1;i<=k;++i){
		ll t=0;
		if (i<=1)	t=b;
		ans=((C*mi%mod-t-b*ans%mod)%mod+mod)%mod*invb%mod;
		C=(n-i+1)%mod*C%mod*inv[i+1]%mod;
	}
	printf("%lld
",ans);
	return 0;
}

如有哪里讲得不是很明白或是有错误，欢迎指正
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