2024.2.11 闲话

[广告位招租]

歌：ひゅーどろ - タケノコ少年 feat. 初音ミク .

求证：

\[\sum_{k=1}^n(-1)^{k+1}\dfrac{\binom{pn}{n-k}}{\binom{pn}nk^p}=\dfrac{H^{(p)}(n)}p \]

一等到底系列 .

\[\begin{aligned}\mathrm{LHS}&=\sum_{k=1}^n(-1)^{k+1}\dfrac{\binom{pn}{n-k}}{\binom{pn}nk^p}\\&=\sum_{k=1}^{n}\dfrac{(-1)^{k+1}}{k^p}\dfrac{\binom{n}{n-k}}{\binom{(p-1)n+k}{(p-1)n}}\\&=\sum_{k=1}^{n-1}\dfrac{(-1)^{k+1}}{k^p}\dfrac{\binom{n}{n-k}}{\binom{(p-1)n+k}{(p-1)n}}+\dfrac{(-1)^{n+1}}{\binom{pn}nn^p}\\&=\sum_{k=1}^{n-1}\dfrac{(-1)^{k+1}}{k^p}\dfrac{n((p-1)n)^{\underline{p-1}}}{(n-k)((p-1)n+k)^{\underline{p-1}}}\dfrac{\binom{n-1}{n-k-1}}{\binom{(p-1)n+k-(p-1)}{(p-1)n-(p-1)}}+\dfrac{(-1)^{n+1}}{\binom{pn}nn^p}\\&={\sum_{k=1}^{n-1}\dfrac{(-1)^{k+1}}{k^p}\dfrac{\binom{n-1}{n-k-1}}{\binom{(p-1)n+k-(p-1)}{(p-1)n-(p-1)}}+\sum_{k=1}^{n-1}\dfrac{(-1)^{k+1}}{k^p}\left(\dfrac{n((p-1)n)^{\underline{p-1}}}{(n-k)((p-1)n+k)^{\underline{p-1}}}-1\right)\dfrac{\binom{n-1}{n-k-1}}{\binom{(p-1)n+k-(p-1)}{(p-1)n-(p-1)}}}{+\dfrac{(-1)^{n+1}}{\binom{pn}nn^p}}\\&=\sum_{k=1}^{n-1}\dfrac{(-1)^{k+1}}{k^p}\dfrac{\binom{n-1}{n-k-1}}{\binom{(p-1)n+k-(p-1)}{(p-1)n-(p-1)}}+\sum_{k=1}^n\dfrac{(-1)^{k+1}}{n^p}\dfrac{\binom{n}{n-k}}{\binom{(p-1)n+k}{(p-1)n}}\\&=\sum_{i=1}^n\sum_{k=1}^i\dfrac{(-1)^{k+1}}{i^p}\dfrac{\binom{i}{i-k}}{\binom{(p-1)i+k}{(p-1)i}}\\&=\sum_{i=1}^n\dfrac1{\binom{(p-1)i+k}{(p-1)i}i^p}\sum_{k=0}^{i-1}(-1)^{n-k+1}\binom ik\\&=\sum_{i=1}^n\dfrac1{pi^p}\\&=\mathrm{RHS}\end{aligned} \]

comic