#HomeWork#21
#Please do not post homework
#Abrar Almahmeed, April 12
#################################


#Question 2:

#I have not started significant work on my project yet.
#But I have thinking about possible directions and plan to begin exploring it soon.


#####################################################################################

#Question 3: 

eknVC := proc(k,n,x)
local f,t,i:
f:=expand(mul(1+x[i]*t,i=1..n)):
coeff(f,t,k):
end:

#Check: 
eknVC(2,4,x);
   x[1] x[2] + x[1] x[3] + x[1] x[4] + x[2] x[3] + x[2] x[4] + x[3] x[4]


ekn(2,4,x);
   x[1] x[2] + x[1] x[3] + x[1] x[4] + x[2] x[3] + x[2] x[4] + x[3] x[4]


evalb(%=%%);
                              true

##########################################################################################

#Question 4: Using the Maple commands "mul", "taylor", and "expand", write a procedure

hkn := proc(k,n,x)
local f,t,i:
f:=expand(taylor(1/mul(1-t*x[i],i=1..n),t=0,k+1)):
f:
end:

#Check:

hkn(2,3,x);
1 + (x[1] + x[2] + x[3]) t

     /    2                               2                   2\ 
   + \x[1]  + x[1] x[2] + x[1] x[3] + x[2]  + x[2] x[3] + x[3] / 

   2    / 3\
  t  + O\t /

#To obtain the coefficient of t^k (eg. in our example t^2) we use the Maple command "coeff"

hkn := proc(k,n,x)
local f,t,i:
f:=expand(taylor(1/mul(1-t*x[i],i=1..n),t=0,k+1)):
coeff(f,t,k):
end:

#Check:
hkn(2,3,x);
       2                               2                   2
   x[1]  + x[1] x[2] + x[1] x[3] + x[2]  + x[2] x[3] + x[3] 


##################################################################################################

#Question 5:

CheckNewton:=proc(k,n)
local LHS,RHS,i:
LHS:=k*ekn(k,n,x):
RHS:=add((-1)^(i-1)*ekn(k-i,n,x)*pkn(i,n,x),i=1..k):
evalb(expand(LHS)=expand(RHS)):
end:

#Check:

CheckNewton(2,4);

                              true


#Run for all 5>=n>=k>=1:

[seq(seq(CheckNewton(k,n),k=1..n),n=1..5)];
[true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]