For the resolved conifold with one outer D-brane in arbitrary framing, we
present some results for the open string partition functions obtained by some
operator manipulations. We prove some conjectures by Aganagic-Vafa and
Aganagic-Klemm-Vafa that relates such invariants to the mirror curve of the
resolved conifold. This establishes local mirror symmetry for the resolved
confolds for holomorphic disc invariants. We also verify an integrality
conjecture of such invariants by Ooguri-Vafa in this case and present closed
formulas for some Ooguri-Vafa type invariants in genus 0 and arbitrary genera