But, in the (19.50) the denominator is
$$
k(k+1)-l(l+1)
$$
that, for ##k=n-1## becomes
$$
(n-1)n-l(l+1)
$$
and, since ##n=l+1## we have
$$
(n-1)n-(n-1)n
$$
Reading the classical Feynman lectures, I encounter the formula(19.53) that gives the radial component of the wave function:
$$
F_{n,l}(\rho)=\frac{e^{-\alpha\rho}}{\rho}\sum_{k=l+1}^n a_k \rho^k
$$
that, for ##n=l+1## becomes
$$
F_{n,l}=\frac{e^{-\rho/n}}{\rho}a_n\rho^n
$$
To find ##a_n## I...