#Please do not post homework
#Caroline Cote, April 13th, 2026, Assignment 21
HelpHW21:= proc(): print(``): end:
read "C21.txt":

#Problem 2: 
#I have not really made any progress on the project yet. I have only tested out some of the DZtools.txt functions that we spoke about in class.

#Problem 3: 

#eknVC(k,n,x): computes ekn the very clever way
eknVC:= proc(k,n,x) local X,t, i:
X:= expand(mul(1 + x[i]*t, i = 1..n)):
coeff(X,t,k):
end:

#Problem 4: 

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

#Problem 5:

#CheckNewton(k,n): 
CheckNewton:= proc(k,n) local i,x, left, right:
left:= k*ekn(k,n,x):
right:= add((-1)^(i-1)*ekn(k-i,n,x)*pkn(i,n,x), i = 1..k):
evalb(expand(left - right) = 0):
end:

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