In financial mathematics involving immunization, I encounter situations where I am trying to calculate
$$(A) \quad v+4v^2+9v^3+ \cdots +n^2v^n=\sum_{t=1}^{n}t^2v^t $$
where $v$ is the present value factor.
I understand that
$$\sum_{t=1}^{n}v^t=a_{\overline{n}\rceil}$$
is the annuity immediate present value factor and
$$\sum_{t=1}^{n}tv^t=(Ia)_{\overline{n}\rceil}=\frac{\ddot{a}_{\overline{n}\rceil}-nv^n}{i}$$
is the increasing annuity immediate present value factor.
Once understood, I find these simplifications very convenient and I tried to come up with a formula for eqn. (A)
Using the fact that
$$\sum_{t=1}^{n}tx^t=\frac{x-x^{n+1}}{(1-x)^2}-\frac{nx^{n+1}}{1-x}$$
taking the derivative on both sides and multiplying $x$ gives me something similar to (A) which I think is
$$(B) \ \quad \sum_{t=1}^{n}t^2x^t=\frac{2x^2(1-x^{n+1}-(n+1)x^{n})}{(1-x)^3}-n(n+1)x^{n+2}-\frac{nx^{n+3}}{1-x}$$
I was simply wondering if there are any known formulae for eqn. (A) using actuarial notation. Trying to deduce something from (B) is quite a tackle for me and I am not confident that I can simplify this.