Question

Coefficient of x^n in the expansion of 1/(1-x)(1-2x)(1-3x) is

Answer 1

Answer:

coefficient of xn in

(1+x+2x2+3x3+.....+nxn)2

My attempt:Let S=1+x+2x2+3x3+...+nxn

xS=x+x2+2x3+3x4+...+nxn+1

(1−x)S=1+x+x2+x3+....+xn−nxn+1−x=1−xn+11−x−nxn+1−x

S=1(1−x)2−x1−x=1−x+x2(1−x)2. (Ignoring terms which have powers of x greater than xn)

So one can say that

coefficient of xn in (1+x+2x2+3x3+.....+nxn)2

=coefficient of xn in (1−x+x2)2(1−x)−4

Is there a shorter way.