[广告位招租]
歌:ひゅーどろ - タケノコ少年 feat. 初音ミク .
求证:
n∑k=1(−1)k+1(pnn−k)(pnn)kp=H(p)(n)p
一等到底系列 .
LHS=n∑k=1(−1)k+1(pnn−k)(pnn)kp=n∑k=1(−1)k+1kp(nn−k)((p−1)n+k(p−1)n)=n−1∑k=1(−1)k+1kp(nn−k)((p−1)n+k(p−1)n)+(−1)n+1(pnn)np=n−1∑k=1(−1)k+1kpn((p−1)n)p−1–––––(n−k)((p−1)n+k)p−1–––––(n−1n−k−1)((p−1)n+k−(p−1)(p−1)n−(p−1))+(−1)n+1(pnn)np=n−1∑k=1(−1)k+1kp(n−1n−k−1)((p−1)n+k−(p−1)(p−1)n−(p−1))+n−1∑k=1(−1)k+1kp(n((p−1)n)p−1–––––(n−k)((p−1)n+k)p−1–––––−1)(n−1n−k−1)((p−1)n+k−(p−1)(p−1)n−(p−1))+(−1)n+1(pnn)np=n−1∑k=1(−1)k+1kp(n−1n−k−1)((p−1)n+k−(p−1)(p−1)n−(p−1))+n∑k=1(−1)k+1np(nn−k)((p−1)n+k(p−1)n)=n∑i=1i∑k=1(−1)k+1ip(ii−k)((p−1)i+k(p−1)i)=n∑i=11((p−1)i+k(p−1)i)ipi−1∑k=0(−1)n−k+1(ik)=n∑i=11pip=RHS
comic


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