0x1+1x2+2x3+......+(n-1)xn=

=0x(0+1)+1x(1+1)+2x(2+1)+........+(n-1)x(n-1+1)

=0x0+0+1x1+1+2x2+2+...........+(n-1)x(n-1)+(n-1)

=[0+1+2+3+.......+(n-1)]+[0x0+1x1+2x2+..........+(n-1)(n-1)]

=(1+n-1)(n-1)/2+(n-1)(n-1+1)[2(n-1)+1]/6

=n(n-1)/2+(n-1)n(2n-1)/6

=n(n-1)(2n-1+3)/6

=n(n-1)^2/3

注：所用公式：

1+2+3........+n=(1+n)n/2

1^2+2^2+3^2+4^2+5^2………………+n^2=n(n+1)(2n+1)/6