query: Does $^nC_{n+1}$ exist?
positive_passages.docid: 155306
positive_passages.text: How are lopsided binomials (eg $\binom{n}{n+1})?$ defined?
negative_passages.docid: 104753
negative_passages.text: $\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?
negative_passages.docid: 53156
negative_passages.text: Infinitely many numbers not expressible in the form $a^n+b^n+c^n$
negative_passages.docid: 35948
negative_passages.text: How to prove $D^n/S^{n-1}\cong S^n$?
negative_passages.docid: 157841
negative_passages.text: Existence of a non-abelian group of order $p^n$.
negative_passages.docid: 69136
negative_passages.text: factorization of a^n+1?
negative_passages.docid: 21782
negative_passages.text: Is $\aleph_0 = \mathbb{N}$?
negative_passages.docid: 83790
negative_passages.text: Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
negative_passages.docid: 143069
negative_passages.text: What are the direct summands in $\mathbb{Z}^n$?
negative_passages.docid: 116361
negative_passages.text: Relationship between prime factorizations of $n$ and $n+1$?
negative_passages.docid: 59782
negative_passages.text: Are there any $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square?
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
negative_passages.docid: 42883
negative_passages.text: Cross product in $\mathbb R^n$
negative_passages.docid: 142347
negative_passages.text: What is the highest power of $n$ in $(n^r-1)!$
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
negative_passages.docid: 77957
negative_passages.text: How is $\mathbb N$ actually defined?
negative_passages.docid: 135808
negative_passages.text: Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$