Here, we compare Buchholz's psi function with normal notations (with Veblen function). I used Rgetar's Ordinal Explorer for the Veblen function in terms of positions of arrays (SVO... etc.)
ψ_0(Ω) = ε_0
ψ_0(Ω^2) = ζ_0
ψ_0(Ω^3) = η_0
ψ_0(Ω^ω) = φ(ω,0)
ψ_0(Ω^Ω) = Γ_0
ψ_0(Ω^Ω+1) = Γ_0*ω
ψ_0(Ω^Ω+2) = Γ_0*ω^2
ψ_0(Ω^Ω+ω) = Γ_0*ω^ω
ψ_0(Ω^Ω+ψ_0(Ω)) = Γ_0*ε_0
ψ_0(Ω^Ω+ψ_0(Ω^2)) = Γ_0*ζ_0
ψ_0(Ω^Ω+ψ_0(Ω^Ω)) = Γ_0^2
ψ_0(Ω^Ω+ψ_0(Ω^Ω+ψ_0(Ω^Ω))) = Γ_0^Γ_0^2
ψ_0(Ω^Ω+Ω) = ε_(Γ_0+1)
ψ_0(Ω^Ω+Ω^2) = ζ_(Γ_0+1)
ψ_0(Ω^Ω+Ω^ω) = φ(ω,Γ_0+1)
ψ_0(Ω^Ω+Ω^ψ_0(Ω^Ω)) = φ(Γ_0,1)
ψ_0(Ω^Ω+Ω^ψ_0(Ω^ψ_0(Ω^Ω))) = φ(φ(Γ_0,1),0)
ψ_0(Ω^Ω*2) = Γ_1
ψ_0(Ω^Ω*ω) = Γ_ω
ψ_0(Ω^Ω*ψ_0(Ω)) = Γ_ε_0
ψ_0(Ω^Ω*ψ_0(Ω^2)) = Γ_ζ_0
ψ_0(Ω^Ω*ψ_0(Ω^ω)) = Γ_φ(ω,0)
ψ_0(Ω^Ω*ψ_0(Ω^Ω)) = Γ_Γ_0
ψ_0(Ω^Ω*ψ_0(Ω^Ω*ψ_0(Ω^Ω))) = Γ_Γ_Γ_0
ψ_0(Ω^(Ω+1)) = φ(1,1,0)
ψ_0(Ω^(Ω+1)+1) = φ(1,1,0)*ω
ψ_0(Ω^(Ω+1)+ψ_0(Ω)) = φ(1,1,0)*ε_0
ψ_0(Ω^(Ω+1)+ψ_0(Ω^Ω)) = φ(1,1,0)*Γ_0
ψ_0(Ω^(Ω+1)+ψ_0(Ω^(Ω+1))) = φ(1,1,0)^2
ψ_0(Ω^(Ω+1)+Ω) = ε_(φ(1,1,0)+1)
ψ_0(Ω^(Ω+1)+Ω^2) = ζ_(φ(1,1,0)+1)
ψ_0(Ω^(Ω+1)+Ω^Ω) = Γ_(φ(1,1,0)+1)
ψ_0(Ω^(Ω+1)*2) = φ(1,1,1)
ψ_0(Ω^(Ω+2)) = φ(1,2,0)
ψ_0(Ω^(Ω+ω)) = φ(1,ω,0)
ψ_0(Ω^(Ω+ψ_0(Ω^Ω))) = φ(1,Γ_0,0)
ψ_0(Ω^(Ω+ψ_0(Ω^(Ω+ψ_0(Ω^Ω)))) = φ(1,φ(1,Γ_0,0),0)
ψ_0(Ω^(Ω2)) = φ(2,0,0)
ψ_0(Ω^(Ω3)) = φ(3,0,0)
ψ_0(Ω^(Ωω)) = φ(ω,0,0)
ψ_0(Ω^(Ωψ_0(Ω))) = φ(ε_0,0,0)
ψ_0(Ω^(Ωψ_0(Ω^Ω))) = φ(Γ_0,0,0)
ψ_0(Ω^Ω^2) = φ(1,0,0,0)
ψ_0(Ω^Ω^2+1) = φ(1,0,0,0)*ω
ψ_0(Ω^Ω^2+ψ_0(Ω)) = φ(1,0,0,0)*ε_0
ψ_0(Ω^Ω^2+ψ_0(Ω^Ω)) = φ(1,0,0,0)*Γ_0
ψ_0(Ω^Ω^2+ψ_0(Ω^Ω^2)) = φ(1,0,0,0)^2
ψ_0(Ω^Ω^2+Ω) = ε_(φ(1,0,0,0)+1)
ψ_0(Ω^Ω^2+Ω^2) = ζ_(φ(1,0,0,0)+1)
ψ_0(Ω^Ω^2*2) = φ(1,0,0,1)
ψ_0(Ω^Ω^2*ω) = φ(1,0,0,ω)
ψ_0(Ω^Ω^2*ψ_0(Ω^Ω^2)) = φ(1,0,0,φ(1,0,0,0))
ψ_0(Ω^(Ω^2+1)) = φ(1,0,1,0)
ψ_0(Ω^(Ω^2+Ω)) = φ(1,1,0,0)
ψ_0(Ω^(Ω^2*2)) = φ(2,0,0,0)
ψ_0(Ω^(Ω^2*3)) = φ(3,0,0,0)
ψ_0(Ω^(Ω^2*ω)) = φ(ω,0,0,0)
ψ_0(Ω^(Ω^2*ψ_0(Ω))) = φ(ε_0,0,0,0)
ψ_0(Ω^(Ω^2*ψ_0(Ω^Ω))) = φ(Γ_0,0,0,0)
ψ_0(Ω^(Ω^2*ψ_0(Ω^Ω^2))) = φ(φ(1,0,0,0),0,0,0)
ψ_0(Ω^Ω^3) = φ(1,0,0,0,0)
ψ_0(Ω^Ω^4) = φ(1,0,0,0,0,0)
ψ_0(Ω^Ω^ω) = φ(1 @^ω) = SVO
ψ_0(Ω^Ω^ψ_0(Ω)) = φ(1 @^ε_0)
ψ_0(Ω^Ω^ψ_0(Ω^2)) = φ(1 @^ζ_0)
ψ_0(Ω^Ω^ψ_0(Ω^Ω)) = φ(1 @^Γ_0)
ψ_0(Ω^Ω^ψ_0(Ω^Ω^2)) = φ(1 @^φ(1,0,0,0))
ψ_0(Ω^Ω^ψ_0(Ω^Ω^ω)) = φ(1 @^φ(1 @^ω))
ψ_0(Ω^Ω^Ω) = φ(1 @^(1,0)) = LVO
ψ_0(Ω^(Ω^Ω+1)) = φ(1 @^(1,0)1,0)
ψ_0(Ω^(Ω^Ω+Ω)) = φ(1 @^(1,0)1,0,0)
ψ_0(Ω^(Ω^Ω+Ω^2)) = φ(1 @^(1,0)1,0,0,0)
ψ_0(Ω^(Ω^Ω+Ω^ω)) = φ(1 @^(1,0) 1 @^ω)
ψ_0(Ω^(Ω^Ω*2)) = φ(2 @^(1,0))
ψ_0(Ω^(Ω^Ω*ω)) = φ(ω @^(1,0))
ψ_0(Ω^(Ω^Ω*ψ_0(Ω))) = φ(ε_0 @^(1,0))
ψ_0(Ω^(Ω^Ω*ψ_0(Ω^Ω))) = φ(Γ_0 @^(1,0))
ψ_0(Ω^(Ω^Ω*ψ_0(Ω^Ω^2))) = φ(φ(1,0,0,0) @^(1,0))
ψ_0(Ω^(Ω^Ω*ψ_0(Ω^Ω^ω))) = φ(φ(1 @^ω) @^(1,0))