由二项式定理,C(n,0)+C(n,1)x+C(n,2)x^2+.+C(n,n)x^n=(1+x)^n,
上式两边对x求导,得:C(n,1)+2C(n,2)x+.+nC(n,n)x^n-1=n(1+x)^n-1
两边同乘以x,得：C(n,1)x+2C(n,2)x^2+.+nC(n,n)x^n=nx(1+x)^n-1
上式两边对x 求导,得:C(n,1)+2^2C(n,2)x+.+n^2C(n,n)x^n-1=n(1+x)^n-1+n(n-1)x(1+x)^n-2,令x=1,得：C(n,1)+2^2C(n,2)+.+n^2C(n,n)=
n(n+1)2^n-2.