#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
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1. Write the above proof in your own words.
A1. 

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2.What is the OEIS A-number of this sequence? TreeSeq(10)
A2. Yes. A-number := A000272

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3. Is this sequence (the number of labelled connected graphs with as many as vertices) on the OEIS 
 A- number?
A3. Yes. A-number := A057500

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4. What is the smallest r such that ATreeSeq(30,r) is not in the OEIS?
A4. r = 13 is the smallest r that is not there in OEIS

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5. What is the ratio of the time it takes between time(TreeSeq(75)) and time(TreeSeq1(75)) ?
A5. time(TreeSeq(75));
                             62.859

time(TreeSeq1(75));
                             1.328

%/`%%`;
                         0.02112664853

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6. Let r_e(n) be the number of rooted trees where every vertex has an even number of children. Set 
up a functional equation for the egf r_e(n)
R_e(x) = Sum(r_e(n)*x^n/n!, n=0.. infinity). What are the first 20 terms of this sequence?
Is it in OEIS?

    7
   / \
  5   9

A6. 

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7. How many labelled trees are there with 27 vertices and 6 leaves?
A7. coeff(TreeSeqL(27, t)[27], t, 6);
              147591129548061384922325901312000000

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8. Conjecture an explicit formula for the average number of leaves of a labelled tree with 
n vertices. 
A8.  

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