#ATTENDANCE QUIZ FOR LECTURE 22 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p22
#with an attachment called
#p22FirstLast.txt
#(e.g. p22DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. Dec. 1, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 8

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER


# Question 1:  Write the above proof (if possible with more details) in your own words. 

# Answer:
# Let's prove this by induction. Our base case is that we have a tree with n=1 verticies with n-1=0 edges.
# This satisfies what we are trying to prove. Our inductive hypothesis is that our statement holds for
# k=n verticies and k-1 edges. Now, if there are no cycles, there must be at least one vertex
# that has only one neighbor. Otherwise, we are forced to have a cycle. Now, if we remove that one edge,
# we now have k=1 = n-1 verticies and k-2 edges. Since we previously did not have a cycle, we still do not
# have a cycle, and we are left with our statement: n-1 verticies and n-2 edges. 

# Question 2: What is the OEIS A-Number of this sequence([1,1,3,16,125,1296,16807...])

# Answer: A000272

# Question 3: Is this sequence ([0,0,1,15,222,3660, 68295...]) in the OEIS? If yes, what is the A-Number?

# Answer: A057500

# Question 4: What is the smallest r such that ATreeSeq(30,r) is not in the OEIS? Optional: Submit it! 

# Answer: r=13 is the smallest. 

# Question 5: What is the ratio of the time it takes between time(TreeSeq(75)); and time(TreeSeq1(75));? 

# Answer: time(TreeSeq(75))/time(TreeSeq1(75)) = 95.63276836

# Question 6: Let r_e(n) be the number of roots trees where every vertex has an even number of children.
# Set up a functional equation for egf of r_e(n). What are the first 20 terms of this sequence? 
# Is it in the OEIS?

# Answer: 
# L := FunEqToSeq(cosh(z), z, 10)
# seq(L[i]*i!, i=1..40) = 
# 1, 0, 3, 0, 65, 0, 3787, 0, 427905, 0, 79549811, 0, 22036379521, 0, 8513206310715, 0, 4374455745966593, 0, 2885264091484122979, 0, 2376040584184726335681, 0, 2389484304129542889498923, 0, 2881763610489447544905661825, 0, 4105338427962827177938910410707, 0, 6820519958449287654130653696838145, 0, 13070687141424899699549462577273867931, 0, 28619427570430844262538330338027857930241, 0, 71005182482850338748897007195214844632176835, 0, 198150708380321732347865616391861363470040195393, 0, 617933503166394591859301761175171448449220698313099, 0
# It is A036778 in the OEIS.

# Question 7: How many labelled trees are there with 27 vertices and 6 leaves? 

# Answer: 
# f:=TreeSeqL(27,t);
# coeff(f[27],t,6) = 5466338131409680923049107456000000 trees

# Question 8: CHALLENGE: Conjecture an explicit formula for the average number of 
# leaves of a labelled tree with n vertices. 
# A bigger challenge: What is the limit if you divide by n
# An even bigger challenge: Prove the conjecture.