#OK to post homework
#Sam Minkin, 10/18, Assignment 12

QUESTION #1:

CPg:=proc(L) local S,l,i:

if nops(L) < 2 then
	RETRURN([]):
fi:

S:=CP(L[1],L[2]):
l:=nops(L):

for i from 3 to l do
	S:=CP(S,L[i]):
od:

S:

end:


QUESTION #2:

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

kthMomentGF:=proc(f,x,k) local x,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:


QUESTION #3:

ScaledMomentGF:=proc(f,x,k) local mk,m2:

mk:=kthMomentGF(f,x,k):
m2:=kthMomentGF(f,x,2):

mk/(m2)^(k/2):

end:


QUESTION #4:

Running seq(ScaledMomentGF(((x + 1)/2)^n, x, k), k = 2 .. 10) we get:
1, 0, 16*(3/16*n^2 - 1/8*n)/n^2, 0, 64*(1/4*n - 15/32*n^2 + 15/64*n^3)/n^3, 0, 256*(-17/16*n + 147/64*n^2 - 105/64*n^3 + 105/256*n^4)/n^4, 0, 1024*(31/4*n - 1185/64*n^2 + 4095/256*n^3 + 945/1024*n^5 - 1575/256*n^4)/n^5

Now, running seq(limit(ScaledMomentGF(((x + 1)/2)^n, x, k), n = infinity), k = 2 .. 10) we get:
1, 0, 3, 0, 15, 0, 105, 0, 945

The A-number in the OEIS: A123023
This sequence gives the recurrence: a(n) = (n-1)*a(n-2), a(0)=1, a(1)=0