51nod 1739

题意

\[\sum\limits_{i=0}^n {n\choose i}|i^k-(n-i)^k| \]

数据范围

做法

\(i=am+b\)
\(|i^k-(n-i)^k|=i^k-(n-i)^k\Longrightarrow am+b<n-am-b\Longrightarrow a<\left\lfloor\frac{n}{m}\right\rfloor -a+\frac{n\%m}{m}-\frac{2b}{m}\Longrightarrow a<\left\lfloor\frac{n}{m}\right\rfloor -a\)
\(|i^k-(n-i)^k|=(n\%m-b)^k-b^k\)

然后相反情况：\(i=(\left\lfloor\frac{n}{m}\right\rfloor -a)\cdot m+b\)，\(|i^k-(n-i)^k|=b^k-(n\%m-b)^k\)
故\(\left\lfloor\frac{n}{m}\right\rfloor~is~odd\)，\(ans=0\)

\(\left\lfloor\frac{n}{m}\right\rfloor~is~even\)
\(Ans={\left\lfloor\frac{n}{m}\right\rfloor\choose \left\lfloor\frac{n}{2\cdot m}\right\rfloor}\sum\limits_{i=0}^{n\%m}{n\%m\choose i}|i^k-(n\%m-i)^k|\)

然后前面那个组合数fft加速转化进制就好了

\(O(m+log_m^n (log_m^n)^2)\)，用的ntt+crt，科技树没点够，后面十个点卡不过去了...