  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.
