query: Proving a Binomial Summation with Induction $\sum_{k=0}^n(-1)^k \binom nk^2$
positive_passages.docid: 143047
positive_passages.text: find $\sum_{k=0}^{t}(-1)^k\binom{t}{k}^2$ for odd t then for even t
negative_passages.docid: 79124
negative_passages.text: Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
negative_passages.docid: 17714
negative_passages.text: Proof $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} = H_n$ by induction
negative_passages.docid: 84010
negative_passages.text: Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
negative_passages.docid: 60860
negative_passages.text: Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
negative_passages.docid: 145048
negative_passages.text: Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
negative_passages.docid: 155769
negative_passages.text: Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
negative_passages.docid: 52113
negative_passages.text: Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
negative_passages.docid: 63510
negative_passages.text: Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
negative_passages.docid: 122006
negative_passages.text: Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 18301
negative_passages.text: Proof by Induction involving Inequality and Factorials as denominators
negative_passages.docid: 130223
negative_passages.text: Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
negative_passages.docid: 5272
negative_passages.text: Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
negative_passages.docid: 59252
negative_passages.text: Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
negative_passages.docid: 125712
negative_passages.text: Working with binomial coefficient $\sum_{k=0}^n (-1)^k \binom nk=0$
negative_passages.docid: 7083
negative_passages.text: Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$