Question

Find a compact form for generating functions of the sequence 1, 8,27,... , k^3

Answer 1

This sequence has generating function

`F(x)=\displaystyle\sum_(k\ge0)k^3x^k`

(if we include
`k=0` for a moment)

Recall that for
`|x|<1`, we have

`\displaystyle\frac1{1-x}=\sum_(k\ge0)x^k`

Take the derivative to get

`\displaystyle\frac1{(1-x)^2}=\sum_(k\ge0)kx^(k-1)=\frac1x\sum_(k\ge0)kx^k`

`\implies\frac x{(1-x)^2}=\displaystyle\sum_(k\ge0)kx^k`

Take the derivative again:

`\displaystyle((1-x)^2+2x(1-x))/((1-x)^4)=\sum_(k\ge0)k^2x^(k-1)=\frac1x\sum_(k\ge0)k^2x^k`

`\implies\displaystyle(x+x^2)/((1-x)^3)=\sum_(k\ge0)k^2x^k`

Take the derivative one more time:

`\displaystyle((1+2x)(1-x)^3+3(x+x^2)(1-x)^2)/((1-x)^6)=\sum_(k\ge0)k^3x^(k-1)=\frac1x\sum_(k\ge0)k^3x^k`

`\implies\displaystyle(x+4x^3+x^3)/((1-x)^4)=\sum_(k\ge0)k^3x^k`

so we have

`\boxed{F(x)=(x+4x^3+x^3)/((1-x)^4)}`