#Nathan Fox
#Homework 25
#I give permission for this file to be posted online

##Read old files
read(`C25.txt`):

#Help procedure
Help:=proc():
print(` problem1(T) `):
end:

##Problem 1
problem1:=proc(K,T) local f,q:
    f:=add(q^i/(i!),i=0..K):
    print(prodmake(f,q,T)):
end:

#Conjecture (OEIS-assisted):
#with(numtheory):
#exp(q)=product((1-q^i)^(-mobius(i)/i),i=1..infinity):
#Proof can be found in hw25.pdf

##Problem 2
#seq(nops(vecptns(n)),n=0..10);
#returns 1, 3, 8, 19, 41, 83, 161, 299, 538, 942, 1610
#This is sequence A182818: G.f.: exp( Sum_{n>=1} sigma(2n)*x^n/n ).
#in OEIS

#I don't know how to prove that the generating functions match, but
#this OEIS entry is closely related to the partition generating
#function