How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$?
36786
Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
126888
How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
87479
Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
146762
Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
108401
Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$
79814
Calculate sums of inverses of binomial coefficients
70248
Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
146533
Having problems proving $ \sum_{r=0}^{n}(r+1) \binom{n}{r} = (n+2)2^{n-1} $
104704
Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially
127150
Finding general formula of $\sum_{i=0}^n i^2 \binom{n}{i}$
143365
Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
3798
Finding formula for sum $S_n = \sum_{i=1}^n i(i-1)$
36361
How to calculate the summation $(\sum_{p = k}^{n} \binom{n}{p}) / 2^n$ quickly?
482
Express $1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$ in a simplifed form
45862
Nested sum of binomial coefficient in closed form
10222
Simplify $\sum (x_i- \mu)^2$
119107
Is it true that $\sum_{k=3}^{n}\binom{n}{k}\binom{k}{3} = \binom{n}{3}\binom{n-3}{k-3}$?