#OK to post
# Soham Palande, Assignment 12, 10/18/2020


#PART 1

Cpg:=proc(L) local temp,Result:
Result:=L:
while nops(Result)>1 do
 temp:=Result[1]:
 temp:=CP(Result[1],Result[2]):
 Result:=subsop(1=temp,Result):
 Result:=subsop(2=NULL,Result):

od:

Result:
end: 


#EXAMPLE

L1:=[1,2,3,4]
L2:=[5,6,7,8]
L3 := [9, 10, 11, 12]

CP(L1,L2)
[6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12]

L:=[L1,L2,L3]
	L := [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]]

Cpg(L)
[[15, 16, 17, 18, 16, 17, 18, 19, 17, 18, 19, 20, 18, 19, 20, 21, 16, 17, 18, 19, 17, 18, 19, 20, 18, 19, 20, 21, 19, 20, 21, 22, 17, 18, 19, 20, 18, 19, 20, 21, 19, 20, 21, 22, 20, 21, 22, 23, 18, 19, 20, 21, 19, 20, 21, 22, 20, 21, 22, 23, 21, 22, 23, 24]]




#PART 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:=AveClever(L):
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:


# EXAMPLE

L:=[4,7,6,8,9]

f:=WtEn(L,x)
f := x^9+x^8+x^7+x^6+x^4

AveClever(L)
34/5

AveGF(f,x)
34/5

kthMomentClever(L,2)
74/25

kthMomentGF(f,x,2)
74/25


#PART 3

#using function kthMomentGF(f,x,k) written in question 2 of the hw

ScaledMomentGF:=proc(f,x,k) local m_k,m_2,denom,res:
m_k:=kthMomentGF(f,x,k):
m_2:=kthMomentGF(f,x,2):
denom:=m_2^(k/2):
res:=m_k/denom:
res:
end:


#PART 4

seq(limit(ScaledMomentGF(((1+x)/2)^n,x,k),n=infinity),k=2..10)
#Yes the sequence is in the OEIS.