#OK to post homework 
#Zhizhang Deng, 10/23/2020, Assignment 12

1. 
    CPg := proc(L) local first, second, n, ans: 
        # try every two combination for elements in L then pass it to CP and join the final answer
        ans := []; 
        n := nops(L);

        for first from 1 to n - 1 by 1 do
            for second from first + 1 to n by 1 do
                # (first, second) is every unordered combination 
                ans := [op(ans), op(CP(L[first], L[second]))]
            end do;
        end do;

        return ans;
    end:

2. 
    AveGF := proc(f, x): 
        return 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) :
        return kthMomentGF(f, x, k) / kthMomentGF(f, x, 2) ^ (k / 2):
    end:

4. 
    sgf := [seq(ScaledMomentGF(((1 + x)/2)^n, x, k), k = 2 .. 10)]
    # sgf now contain a list of element with some of them have n in it
    seq(limit(sgf[i], n = infinity), i = 1 .. nops(sgf))
    = 1, 0, 3, 0, 15, 0, 105, 0, 945

    this sequence is in OEIS, with A number: A123023