Is $T(1)=1?$

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$T\left ( n \right )=2^{n}T\left ( \frac{n}{2} \right )+n^{n}$

$=2^{n}\left ( 2^{n/2}T\left ( \frac{n}{2^{2}} \right )+\left ( \frac{n}{2} \right ) ^{n/2}\right )+n^{n}$

$=2^{n+\frac{n}{2}+\frac{n}{4}+........}T\left ( 1 \right )+n^{n}+\left ( \frac{n}{2} \right )^{n/2}+\left ( \frac{n}{4} \right )^{n/4}+.......1$

$=2^{n}+n^{n}+............$

$=O(n^{n})$

$=2^{n}\left ( 2^{n/2}T\left ( \frac{n}{2^{2}} \right )+\left ( \frac{n}{2} \right ) ^{n/2}\right )+n^{n}$

$=2^{n+\frac{n}{2}+\frac{n}{4}+........}T\left ( 1 \right )+n^{n}+\left ( \frac{n}{2} \right )^{n/2}+\left ( \frac{n}{4} \right )^{n/4}+.......1$

$=2^{n}+n^{n}+............$

$=O(n^{n})$

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