Nonnegative hyperplane twist cohomology by restriction ($h^0({\mathbf{P}}^n,\mathcal{O}(m))=\left(\genfrac{}{}{0ex}{}{n+m}{n}\right),H^{r>0}=0$)

For $m\ge 0$, these sheaf cohomology dimensions follow from the vanishing for $\mathcal{O}$ and the hyperplane exact sequence for twisting sheaves. Induct on dimension and then on $m$ in $0\to \mathcal{O}(m-1)\to \mathcal{O}(m)\to i_{\ast }{\mathcal{O}}_{{\mathbf{P}}^{n-1}}(m)\to 0$. The long exact sequence in sheaf cohomology gives higher-degree vanishing and the recurrence $h_n^0(m)-h_n^0(m-1)=\left(\genfrac{}{}{0ex}{}{n-1+m}{n-1}\right)$. Starting with $h_n^0(0)=1$ yields the formula; the corresponding homogeneous monomials form a basis of global sections.