求1^2Cn1+2^2Cn2+3^2Cn3+.+n^2Cnn的值

Question

求1^2Cn1+2^2Cn2+3^2Cn3+.+n^2Cnn的值

Answer 1

consider
(1+x)^n = C(n,0) +C(n,1)x+...+C(n,n)x^n
n(1+x)^(n-1) = C(n,1)+2C(n,2)x+....+nC(n,n)x^(n-1)
nx(1+x)^(n-1) =C(n,1)x+2C(n,2)x^2+....+nC(n,n)x^n
n[(1+x)^(n-1) + (n-1)x(1+x)^(n-2)] =C(n,1)+2^2.C(n,2)+.....+n^2.C(n,n)x^(n-1)
put x=1
n[2^(n-1) + (n-1)2^(n-2)] =1^2.C(n,1)+2^2.C(n,2)+.....+n^2.C(n,n)
1^2.C(n,1)+2^2.C(n,2)+.....+n^2.C(n,n) = n[2^(n-1) + (n-1)2^(n-2)]
=n(n+1).2^(n-2)