LUOGU P4345 SHOI2015超能粒子炮·改题解

$$f(n,k)=\sum _{i=0}^k{C}_n^i\%p$$
$$f(n,k)=\sum _{i=0}^k{C}_{\frac{n}{p}}^{\frac{i}{p}}{C}_{n\%p}^{i\%p}\%p$$
$$=((\sum _{i=0}^{\frac{k}{p}-1}{C}_{\frac{n}{p}}^i\sum _{j=0}^{p-1}{C}_{n\%p}^j)+(C_{\frac{n}{p}}^{\frac{k}{p}}\sum _{i=0}^{k\%p}{C}_{n\%p}^i))\%p$$
$$=(f(\frac{n}{p},\frac{k}{p}-1)f(n\%p,p-1)+C_{\frac{n}{p}}^{\frac{k}{p}}f(n\%p,k\%p))\%p$$