  Two formulas for the classical number $C_n$ of lines on a generic
hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from
the formulas by Dominici, Harris, Libgober, and van der Waerden-Zagier. We
review the splitting principle computation by Harris obtaining a similar
general closed-form formula in terms of the Catalan numbers and elementary
symmetric polynomials. This in turn yields $C_n$ as a linear difference
recursion relation of unbounded order. Thus, for the sequence of certain linear
combinations of $C_n$, a simple generating function is found. Then, a result
from random algebraic geometry by Basu, Lerario, Lundberg, and Peterson, that
expresses these classical enumerative invariants as proportional to the
Bombieri norm of particular polynomial determinants, yields another
combinatorial expansion in terms of certain set compositions and block labeling
counting. As an example, we compute this combinatorial interpretation for the
cases of 27 lines on a cubic surface and 2875 lines on a quintic threefold. As
an application, we reobtain the parity and asymptotic upper bound of the
sequence. In an appendix, we generalize the splitting principle calculation to
obtain a formula for the number of lines on a generic complete intersection.
