query: Simplify the expression - sum of squares of binomial coefficients
positive_passages.docid: 1125
positive_passages.text: Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
negative_passages.docid: 133420
negative_passages.text: Computing the binomial series: $\binom{n}{2}\cdot2^2+\binom{n}{4}\cdot2^4+\binom{n}{6}\cdot2^6+\cdots+ \binom{n}{n}\cdot2^n$
negative_passages.docid: 58972
negative_passages.text: Proof/Derivation of Closed form of Binomial Expression $\sum\limits_{k=0}^{2n}(-1)^k\binom{2n}{k}^2$
negative_passages.docid: 127150
negative_passages.text: Finding general formula of $\sum_{i=0}^n i^2 \binom{n}{i}$
negative_passages.docid: 39957
negative_passages.text: Product of binomial coefficient as a basis
negative_passages.docid: 15403
negative_passages.text: Sum of square of function
negative_passages.docid: 150355
negative_passages.text: Finding sum of binomial series
negative_passages.docid: 79814
negative_passages.text: Calculate sums of inverses of binomial coefficients
negative_passages.docid: 94902
negative_passages.text: Simplifying a geometric sum
negative_passages.docid: 65983
negative_passages.text: Combinatorial proof of binomial coefficient summation identity
negative_passages.docid: 20443
negative_passages.text: Calculate sum of squares of first $n$ odd numbers
negative_passages.docid: 19776
negative_passages.text: Simplify a combinatorial sum
negative_passages.docid: 70248
negative_passages.text: Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
negative_passages.docid: 110441
negative_passages.text: Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
negative_passages.docid: 118024
negative_passages.text: Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$
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: 151757
negative_passages.text: Simplify $\sum_{i=0}^{N/2}{N-i\choose i}$