closed form for combinatorial sum
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
98941
Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $?
39659
Is there a closed formula for this summation?
49771
Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.
125981
Closed-Form Solution to Infinite Sum
65983
Combinatorial proof of binomial coefficient summation identity
35855
Closed form for an infinite product
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
25826
Proof of a combinatorial identity: $\sum\limits_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$
151085
Simplify the sum $ \sum_{i=0}^{k}(-1)^i i \binom{n}{i} \binom{n}{k-i}$
116722
Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$
119606
Closed form of summation
141390
Closed form for $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2+a^2}$
21000
Calculating $\sum_{k=1}^nk(k!)$ combinatorially
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
45862
Nested sum of binomial coefficient in closed form
47753
Closed form of series with factorial-squared denominator?