Here by $P^n$ I mean $CP^n$, and what I want to do is to calculate the number of global sections of the holomorphic tangent bundle of $CP^n$.

If $n=1$, it is well known that $h^0(P^1, TP^1)=h^o(P^1,\mathcal{O}_{P^1}(2))=3$.

If $n>1$, I did some calculation in local coordinates, and find out that $h^0(P^n, TP^n) = n(n+1)$.

I am not sure if this is the correct answer and wonder if anyone else has calculated this before.

Besides, does anybody know the value of $h^1(P^n, TP^n)$? Even the $n=2$ case is enough for me. Many thanks!