#OK to post homework
#Kent Mei, 10/18/20, Assignment 12

#---------------------------------
#Part 1

CPg:=proc(L) local n, Ans, i:

n := nops(L):
if n < 1 then 
 RETURN []:
fi:

Ans := L[1]:

for i from 2 to n do
 Ans := CP(Ans, L[i]):
od:

Ans:

end:


#---------------------------------
#Part 2

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

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

n := add(coeffs(expand(f))):
subs(x=1,f1)/n:

end:


#---------------------------------
#Part 3

ScaledMomentGF:=proc(f,x,k):

kthMomentGF(f,x,k)/kthMomentGF(f,x,2)^(k/2):

end:


#---------------------------------
#Part 4

#k =  2: 1
#k =  3: 0
#k =  4: 16*((3/16)*n^2-(1/8)*n)/n^2
#k =  5: 0
#k =  6: 64*((1/4)*n-(15/32)*n^2+(15/64)*n^3)/n^3
#k =  7: 0
#k =  8: 256*(-(17/16)*n+(147/64)*n^2-(105/64)*n^3+(105/256)*n^4)/n^4
#k =  9: 0
#k = 10: 1024*((31/4)*n-(1185/64)*n^2+(4095/256)*n^3-(1575/256)*n^4+(945/1024)*n^5)/n^5


#[seq(limit(ScaledMomentGF(((1+x)/2)^n,x,k), n = infinity), k = 2..10)]
#[1,0,3,0,15,0,105,0,945]
#It is in the OEIS as A123023.