  The building blocks of 6d $(1,0)$ SCFTs include certain rank one theories
with gauge group $G=SU(3),SO(8),F_4,E_{6,7,8}$. In this paper, we propose a
universal recursion formula for the elliptic genera of all such theories. This
formula is solved from the elliptic blowup equations introduced in our previous
paper. We explicitly compute the elliptic genera and refined BPS invariants,
which recover all previous results from topological string theory, modular
bootstrap, Hilbert series, 2d quiver gauge theories and 4d $\mathcal{N}=2$
superconformal $H_{G}$ theories. We also observe an intriguing relation between
the $k$-string elliptic genus and the Schur indices of rank $k$ $H_{G}$ SCFTs,
as a generalization of Lockhart-Zotto's conjecture at the rank one cases. In a
subsequent paper, we deal with all other non-Higgsable clusters with matters.
