#OK to post homework
#William Wang, 10/18/2020, Assignment 12

#1.
CPg := proc(L) local i, j, k, l, S:
S := []:
for i to nops(L) do
 for j from 2 to nops(L) do
  if i < j then
   S := [op(S), op(CP(L[i], L[j]))]:
  fi:
 od:
od:
S:
end:

CPg([[1, 2], [3, 4]]);
                          [4, 5, 5, 6]

CPg([[3, 4], [5, 6]]);
                         [8, 9, 9, 10]

CPg([[1, 2, 3, 4], [5, 6, 7, 8]]);
     [6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12]

CPg([[1, 2], [3, 4], [5, 6]]);
             [4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10]





#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 to k do
 f1 := expand(x*diff(f1, x)):
od:
subs(x = 1, f1):
end:

L := [3, 4, 5, 6, 7]

#Check that AveGF(WtEn(L, x), x) = AveClever(L)
evalb(AveGF(WtEn(L, x), x) = AveClever(L));
                              true

#Check that kthMomentGF(WtEn(L, x), x, k) = kthMomentClever(L, k)
seq(evalb(kthMomentGF(WtEn(L, x), x, k) = kthMomentClever(L, k)), k = 1 .. 4);
                     true, true, true, true





#3.
ScaledMomentGF := proc(f, x, k):
kthMomentGF(f, x, k)/kthMomentGF(f, x, 2)^(1/2*k):
end:

#Checking that ScaledMomentGF works for L (as defined above in #2) and k=4:
kthMomentGF(WtEn(L, x), x, 4)/kthMomentGF(WtEn(L, x), x, 2)^2;
                               17
                               --
                               10

ScaledMomentGF(WtEn(L, x), x, 4);
                               17
                               --
                               10





#4.
seq(ScaledMomentGF(((1 + x)/2)^n, x, k), k = 2 .. 10);
            /  1     3   2\        /1     15  3   15  2\     
         16 |- - n + -- n |     64 |- n + -- n  - -- n |     
            \  8     16   /        \4     64      32   /     
   1, 0, ------------------, 0, ------------------------, 0, 
                  2                         3                
                 n                         n                 

         /  17     105  3   147  2   105  4\     
     256 |- -- n - --- n  + --- n  + --- n |     
         \  16     64       64       256   /     
     ---------------------------------------, 0, 
                        4                        
                       n                         

          /31     945   5   4095  3   1185  2   1575  4\
     1024 |-- n + ---- n  + ---- n  - ---- n  - ---- n |
          \4      1024      256        64       256    /
     ---------------------------------------------------
                              5                         
                             n                          

#Limit as n approaches infinity for all k=2..10:
#1,0,3,0,15,0,105,0,945
#This sequence is in the OEIS, its A number is A123023