CF891E
\[\sum\limits_{i = 1}^{k} \frac{(i - 1)!}{n^i} (\sum\limits_{j = 1}^{n} \prod\limits_{k \neq j} (\sum\limits_{t = 0} a_k \frac{x^t}{t!} - x \sum\limits_{t = 0}\frac{x^{t - 1}}{(t - 1)!}) )\sum\limits_{g = 0} \frac{x^g}{g!} [x^{i - 1}]
\]
\[\sum\limits_{i = 1}^{k} \frac{(i - 1)!}{n^i} (\sum\limits_{j = 1}^{n} \prod\limits_{k \neq j} (a_k e^x - x e^x)) e^x [x^{i - 1}]
\]
\[(\sum\limits_{j = 1}^{n} \prod\limits_{k \neq j} (a_k - x)) e^{nx} \sum\limits_{i = 1}^{k} \frac{(i - 1)!}{n^i}[x^{i - 1}]
\]
设这个多项式 \((\sum\limits_{j = 1}^{n} \prod\limits_{k \neq j} (a_k - x))\) 是 \(\sum\limits_{i = 0}^{n} a_i x^i\)
\[\sum\limits_{j = 0}^{n} a_j x^j e^{nx} \sum\limits_{i = 1}^{k} \frac{(i - 1))!}{n^i} [x^{i - 1}]
\]
\[\sum\limits_{j = 0}^{n} a_j \sum\limits_{i = 1}^{k} \frac{(i - 1)!}{n^i} \frac{n^{i - j - 1}}{(i - j - 1)!}
\]
\[\sum\limits_{j = 0}^{n} a_j \frac{1}{n^{j + 1}} \sum\limits_{i = 1}^{k} \frac{(i - 1)!}{(i - j - 1)!}
\]
\[\sum\limits_{j = 0}^{n} a_j \frac{1}{n^{j + 1}} j! \sum\limits_{i = 1}^{k} \binom{i - 1}{i - j - 1}
\]
\[\sum\limits_{j = 0}^{n} a_j \frac{1}{n^{j + 1}} j! \sum\limits_{i = 0}^{k - j - 1} \binom{i + j}{i}
\]
\[\sum\limits_{j = 0}^{n} a_j \frac{1}{n^{j + 1}} j! \binom{k}{j + 1}
\]
