query: Sum of binomial coefficients multiplied by powers of two and something else
positive_passages.docid: 147910
positive_passages.text: Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
negative_passages.docid: 35342
negative_passages.text: Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
negative_passages.docid: 6803
negative_passages.text: Expression for power of a natural number in terms of binomial coefficients
negative_passages.docid: 39957
negative_passages.text: Product of binomial coefficient as a basis
negative_passages.docid: 118372
negative_passages.text: Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
negative_passages.docid: 144571
negative_passages.text: A conjecture including binomial coefficients
negative_passages.docid: 87589
negative_passages.text: Evaluate $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+...+\binom{n}{2k}$
negative_passages.docid: 45672
negative_passages.text: Even/Odd Binomial Coefficients
negative_passages.docid: 83229
negative_passages.text: Nested summations and their relation to binomial coefficients
negative_passages.docid: 115686
negative_passages.text: A finite summation of double binomial coefficients
negative_passages.docid: 19581
negative_passages.text: $\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
negative_passages.docid: 28027
negative_passages.text: Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
negative_passages.docid: 4201
negative_passages.text: Prove $\sum\binom{n}{k}2^k = 3^n$ using the binomial theorem
negative_passages.docid: 61270
negative_passages.text: Binomial Theorem Sum
negative_passages.docid: 151384
negative_passages.text: Summation of series involving binomial coefficients and polynomial of degree at most n-1
negative_passages.docid: 42606
negative_passages.text: Orthogonality for Binomial Coefficients
negative_passages.docid: 79814
negative_passages.text: Calculate sums of inverses of binomial coefficients