Erratum to: Proof of new S-matrix formula from classical solutions in open string field theory: Or, deriving on-shell open string field amplitudes without using Feynman rules, Part II

We study relation between the gauge invariant quantity obtained in [arXiv:1908.09784] and the Feynman diagrams in the dressed $ \mathcal B_0 $ gauge in the open cubic string field theory. We derive a set of recurrence relations that hold among the terms of this gauge invariant quantity. By using these relations, we prove that this gauge invariant quantity equals the $S$-matrix at the tree level. We also present a proof that a set of new Feynman rules proposed in [arXiv:2003.05021 [hep-th]] reproduces the onshell disc amplitudes correctly by using the same combinatorial identities.