Explanation of the following code which taken from mathworld

Question

Can anyone explain to me how does this code function?

AdomianPolynomials[u_, F_, o_] := CoefficientList[
   ExpandAll[
    Series[
     F[
      Sum[λ^k u[k], {k, o}]
      ],
     {λ, 0, o}
     ]
    ], λ]

The original code was get from noteboook of this website http://mathworld.wolfram.com/AdomianPolynomial.html

Answer 1

Sum, evaluates the sum $\sum_{k=0}^{o}{\lambda^k\,u(k)}$

With[{o = 3},
 Sum[λ^k u[k], {k, o}]
 ]

λ u[1] + λ^2 u[2] + λ^3 u[3] 

Series, generates a power series expansion for $F(\sum_{k=0}^{o}{\lambda^k\,u(k)})$ about the point $0$ to order $(\lambda-0)^o$.

With[{o = 3},
 Series[F[Sum[λ^k u[k], {k, o}]], {λ, 0, o}]
 ]

CoefficientList, gives a list of coefficients of powers of $\lambda$ in the previos expression, starting with power 0.

ExpandAll, expands out all products and integer powers in any part of the expression in the argument.

With[{o = 3},
 CoefficientList[
  ExpandAll[
   Series[F[Sum[λ^k u[k], {k, o}]], {λ, 0, 
     o}]], λ]
 ]

Follow the links for the documentation.