  Explicit relations between the effective-range expansion and the nuclear
vertex constant or asymptotic normalization coefficient (ANC) for the virtual
decay $B\to A+a$ are derived for an arbitrary orbital momentum together with
the corresponding location condition for the ($A+a$) bound-state energy. They
are valid both for the charged case and for the neutral case. Combining these
relations with the standard effective-range function up to order six makes it
possible to reduce to two the number of free effective-range parameters if an
ANC value is known from experiment. Values for the scattering length, effective
range, and form parameter are determined in this way for the $^{16}$O+$p$,
$\alpha+t$ and $\alpha+^3$He collisions in partial waves where a bound state
exists by using available ANCs deduced from experiments. The resulting
effective-range expansions for these collisions are valid up to energies larger
5 MeV.
