Properties of C(2n,n)
☀ | C(2n,n) = (2n)! / (n!)2 = 2n *(2n-1)(2n-3)(2n-5)...........1 / n! ( no. of terms in red will be equal to n) |
☗ | C(2n,n) is always an even number. |
☞ | C(2n,n) is the sum of co-efficients in the expansion of (1-x)-n |
☻ | P(n,r) =r! * C(n,r) which implies P(2n,n) = (2n)! / n! or P(2n,n) =n! * C(2n,n) = 2n *(2n-1)(2n-3)(2n-5)...........1 ( no. of terms in red will be equal to n) |
♜ | We take x= floor[(n-1)/2] , then C(2n,n) = 2(x+1) * [(2n-1)(2n-3)....upto (n+1) or (n+2) terms depending whether n is even or odd . In other words Floor (n/2) terms] / (n-x-1)! |
♞ | Since C(n,r) =(n/r)*C(n-1,r-1), C(2n,n) =2*C(2n-1,n-1) and C(2n+1,n+1) = [(2n+1)/(n+1)]*C(2n,n) |
☂ | C(2n+1,n+1)= [(2+1/n)/(1+1/n)]*2C(2n-1,n-1); when n=1, C(2n+1,n+1) = 3*C(2n-1,n-1); As n -->∞ , C(2n+1,n+1) = 4*C(2n-1,n-1); |