In a recent article published in 2017, Barroso, Pérez, and Popescu-Pampu employ the Lagrange inversion formula to solve certain Newton-Puiseux equations when the solutions to the inverse problems are given. More precisely, for an irreducible
f
(
x
,
y
)
∈
K
[
[
x
,
y
]
]
f(x,y)\in K[[x,y]]
over an algebraically closed field
K
K
of characteristic zero, they calculate the coefficients of
η
(
x
1
/
n
)
\eta (x^{1/n})
which would meet
f
(
x
,
η
(
x
1
/
n
)
)
=
0
f(x,\eta (x^{1/n}))=0
in terms of the coefficients of
ξ
(
y
1
/
m
)
\xi (y^{1/m})
that satisfy
f
(
ξ
(
y
1
/
m
)
,
y
)
=
0
f(\xi (y^{1/m}),y)=0
. This article will present an alternative approach to solving the problem using diagonalizations on polynomial sequences of binomial-type. Along the way, a close relationship between binomial-type sequences and the Lagrange inversion formula will be observed. In addition, it will extend the result to give the coefficients of
η
(
x
1
/
n
)
\eta (x^{1/n})
directly in terms of the coefficients of
f
(
x
,
y
)
f(x,y)
. As an application, an infinite series formula for the roots of complex polynomials will be obtained together with a sufficient condition for its convergence.