Solve the recurrence relation $T(n) = nT^2(n/2)$

Question

Solve the recurrence relation $T(n) = nT^2(n/2)$ with initial condition $T(1) = 6$.

So far I believe that I should let $n = 2k$ and then make the substitution for $ak = \log T(2k)$. From here I am stumped, any help is appreciated.

Answer 1

Assuming this is for a computer science class and you've omitted floors/ceilings for brevity, here's a hint: set $U(n)=\lg T(n)$ (or use your favorite base). Take the log of both sides of the recurrence and express the result in terms of $U$; you get $$U(n)=2 U(n/2)+\lg n.$$

Now apply the master theorem to determine $U(n)$, and now you know $T(n)$ since $T(n)=2^{U(n)}.$