/ 2n \         / n \
1. {      }  =  2 {     }  + n^2
    \  2 /         \ 2 /


     n
   _____  
   \        / n \ ^2    / 2n \
2.  \      {     }   = {      }
    /       \ k /       \  n /
   /____
    k=0

     n
   _____  
   \        / 2n \  / 2n-2k \     
3.  \      {      }{         } =  4^n
    /       \ 2k /  \  n-k  /
   /____
    k=0
  

    / nk \         / k \       / n+1 \
4. {      } = n^2 {     } +  k{       }
    \  2 /         \ 2 /       \  2  /


     r
   _____
   \        / n \  /  m  \      / n+m \
5.  \      {     }{       } =  {       }
    /       \ k /  \ r-k /      \  r  /
   /____
    k=0


  
    / n \  / r \      / n \  / n-k \
6. {     }{     } =  {     }{       },    k <= r <= n
    \ r /  \ k /      \ k /  \ r-k /
   


    / n \     / n-3 \      / n-1 \     / n-2 \     / n-3 \
7. {     } - {       } =  {       } + {       } + {       }
    \ k /     \  k  /      \ k-1 /     \ k-1 /     \ k-1 /


     n
   _____
   \       
8.  \    k*k! = (n+1)! - 1
    /      
   /____
    k=1


     n-1
   _____
   \        / n \      /  m  \                
9.  \      {     }k^k {       }(n-k)^(n-k-1) = n^n
    /       \ k /      \ r-k /                 
   /____
    k=0






Here are some involving Fibonacci numbers, i.e. F_0=1, F_1=1, and
F_i = F_(i-1) + F_(i-2).


     n
   _____   ____      ____    
   \      |         |         /|
    \     |__     = |__    -   |
    /     |         |          |
   /____  |         |          | 
    i=0     2i        2n+1


     n
   _____   ____ 2     ____  ____   
   \      |          |     |
    \     |__      = |__   |__
    /     |          |     |
   /____  |          |     | 
    i=0     i           n    n+1



    
   ____   ____       ____ 2     
  |      |          |           
  |__    |__    -   |__     =  ( -1)^n
  |      |          |          
  |      |          |              
    n+1    n-1        n



     2n
   _____          ____       ____        
   \             |          |           /|
    \    (-1)^i  |__     =  |__     -    |
    /            |          |            |
   /____         |          |            |
    i=0             i         2n-1


   ____       ____  ____       ____   ____
  |          |     |          |      |
  |__     =  |__   |__     +  |__    |__
  |          |     |          |      |
  |          |     |          |      |
     n+k        n+1    k         n      k-1

   ____       ____  ____       ____   ____
  |          |     |          |      |
  |__     =  |__   |__     +  |__    |__
  |          |     |          |      |
  |          |     |          |      |
     2n        n     n+1         n      n-1


   ____  |  ____
  |      | | 
  |__    | |__
  |      | |
  |      | |
     n   |    kn



      /  ____    ____ \      ____
     /  |       |      \    |
gcd {   |__     |__     } = |__
     \  |       |      /    |
      \ |     , |     /     |  
           n       m          gcd(n,m)