query: Formula for sum of all binomial coefficients $\binom{n}{k}$ where $k$ is divisible by 4
positive_passages.docid: 49234
positive_passages.text: Sum of binomial coefficients with index divisible by 4
negative_passages.docid: 122024
negative_passages.text: Asymptotic of a sum involving binomial coefficients
negative_passages.docid: 86900
negative_passages.text: Formula for the $1\cdot 2 + 2\cdot 3 + 3\cdot 4+\ldots + n\cdot (n+1)$ sum
negative_passages.docid: 151085
negative_passages.text: Simplify the sum $ \sum_{i=0}^{k}(-1)^i i \binom{n}{i} \binom{n}{k-i}$
negative_passages.docid: 126587
negative_passages.text: Divisiblity of binomial coefficients
negative_passages.docid: 7083
negative_passages.text: Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
negative_passages.docid: 151384
negative_passages.text: Summation of series involving binomial coefficients and polynomial of degree at most n-1
negative_passages.docid: 83400
negative_passages.text: Sum of products of binomial coefficients: $ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $
negative_passages.docid: 132249
negative_passages.text: Sum of every $k$th binomial coefficient.
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 21000
negative_passages.text: Calculating $\sum_{k=1}^nk(k!)$ combinatorially
negative_passages.docid: 5332
negative_passages.text: Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$
negative_passages.docid: 59638
negative_passages.text: How to prove this result on binomial coefficients?
negative_passages.docid: 94784
negative_passages.text: $\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ using Counting argument
negative_passages.docid: 41354
negative_passages.text: Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
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: 79814
negative_passages.text: Calculate sums of inverses of binomial coefficients