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);

 Computation of C(2n,n) by 3 Different Methods n 2n Method-I C(2n,n)=(2n)! / (n!)2 A C(2n-1,n-1) B C(2n+1,n+1) B/A C(2n+1,n+1) / C(2n-1,n-1) 2n (2n-1)(2n-3)(2n-5).........1 n! Method-II C(2n,n)=2n *(2n-1)(2n-3)(2n-5)...........1 / n! x 2(x+1) No. of terms-g (2n-1)=g1 (2n-1)(2n-3)..upto g terms i.e.(n+1) or (n+2) terms (n-x-1)! Method-III C(2n,n) = 2(x+1) * [(2n-1)(2n-3)....upto (n+1) or (n+2) ]/  (n-x-1)! P(2n,n) = n! * C(2n,n)