query: Formula for the coefficients in the expansion of $(1+r+r^2+\cdots +r^k)^n$.
positive_passages.docid: 143264
positive_passages.text: Is there a nice way of expanding multinomials of the form $(1+x+\cdots+x^l)^n$?
negative_passages.docid: 61047
negative_passages.text: Find the coefficient of ${x^9}$ in the expansion of $ (1 + x)( 1 + x^2)( 1 + x^3)..(1 + x^{100})$
negative_passages.docid: 134311
negative_passages.text: How to determine equation for $\sum_{k=1}^n k^3$
negative_passages.docid: 138082
negative_passages.text: What are the coefficients in the expansion of $(x+y)(x+2y) \cdots (x+ny)$?
negative_passages.docid: 30518
negative_passages.text: Formula for $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$?
negative_passages.docid: 39909
negative_passages.text: Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$
negative_passages.docid: 149685
negative_passages.text: Formula for $\sqrt{a-\sqrt{a+\sqrt{a+\ldots}}}$
negative_passages.docid: 36639
negative_passages.text: Prove by induction that $\sum_{r=0}^{n}\begin{pmatrix}n \\ r \end{pmatrix}=2^k$
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 47827
negative_passages.text: Prove that $\sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$
negative_passages.docid: 87479
negative_passages.text: Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
negative_passages.docid: 562
negative_passages.text: Formula for calculating $\sum_{n=0}^{m}nr^n$
negative_passages.docid: 98941
negative_passages.text: Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $?
negative_passages.docid: 7063
negative_passages.text: How to compute the  formula $\sum  \limits_{r=1}^d r \cdot 2^r$?
negative_passages.docid: 57549
negative_passages.text: Finding the sum $\sum_{k=1}^rk^2\binom {n-k}{r-k}$
negative_passages.docid: 82591
negative_passages.text: Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
negative_passages.docid: 64581
negative_passages.text: How to find the sum $\sum\limits_{k=1}^n (k^2+k+1)k!$?