1.
CPg:=proc(L) local temp,i:
temp:=CP(L[1],L[2]):
for i from 3 to nops(L) do
  temp:=CP(temp,L[i]):
od;
temp:
end:

2.
AveGF:=proc(f,x):
subs(x=1,diff(f,x))/subs(x=1,f):
end:

kthMomentGF:=proc(f,x,k) local mu,f1,i:
mu:=AveGF(f,x):
f1:=f/x^mu/subs(x=1,f):
for i from 1 to k do
f1:=expand(x*diff(f1,x)):
od:
subs(x=1,f1):
end:

3.
ScaledMomentGF:=proc(f,x,k):
kthMomentGF(f,x,k)/kthMomentGF(f,x,2)^(k/2):
end:

4.
(1069*n^4-6752*n^3+16984*n^2-19232*n+7936)/n^4
As n goes to infinity, sequence is 1, 0, 3, 0, 15, 0, 105, 0, 945.
It is in oeis A123023.