Expand[Binomial[z+3,z]]

A[ℰ_] := Expand[ℰ /. (Subscript[z, n_]):>FunctionExpand@Binomial[z+n,n]]

Table[A[(Subscript[z, n])], {n,0,10}]

B[1]=1;
B[z]:=(Subscript[z, 1])-1;
B[ℰ_]:=Expand@Sum[Coefficient[ℰ,z,n]*B[(z^n)],{n,0,Exponent[ℰ,z]}];
B[(z^(n_))]:=B[(z^n)]=Expand[n!(Subscript[z, n])+B[(z^n)-n!A[(Subscript[z, n])]]]

B[1]

B[z]//A

B[(z^5)]//A

Table[B[(z^n)], {n, 0,5}]

B[Sum[(((ζ^n)(z^n))/(n!)),{n,0,5}]]

n1=30;
F=B[Sum[(((ζ^n)(z^n))/(n!)),{n,0,n1}]];
Table[Simplify@FindGeneratingFunction[Table[Coefficient[Coefficient[F,(Subscript[z, n])],ζ,k],{k,n-1,n1}],ζ, FunctionSpace -> All],{n,1,2}]

Series[-(E^(-2x))+(E^(-x)), {x,0,10}]

n1=30;
F=B[Sum[(((ζ^n)(z^n))/(n!)),{n,0,n1}]];
Table[
FullSimplify[FindSequenceFunction[Table[k!Coefficient[Coefficient[F,(Subscript[z, n])],ζ,k],{k,1,n1}]][ζ]],
{n,1,7}]