  We derive a recursion relation of the Feynman diagrams of the effective
action for the third Legendre transformation in case of the bosonic field
theory with cubic interaction. We apply the recursion relation to obtain the
Feynman diagrams of the effective action for the third Legendre transformation
up to the five-loop order. The three-particle irreducibility of the Feynman
diagrams of the effective action for the third Legendre transformation is shown
by induction.
