We investigate the bifurcation cascades of a linear librational orbit in a generalized class of Henon-Heiles potentials. The stability traces of the new orbits created at its bifurcations are found numerically to intersect linearly at the saddle energy (e=1), forming what we term the "Henon-Heiles fans." In the limit close to the saddle energy (e -> 1), where the dynamics is nearly chaotic, we derive analytical asymptotic expressions for the stability traces of both types of orbits and confirm the numerically determined properties of the generalized Henon-Heiles fans. As a bonus of our results, we obtain analytical approximations for the bifurcation energies e(n) which become asymptotically exact for e(n)-> 1.