#ATTENDANCE QUIZ FOR LECTURE 19 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p19
#with an attachment called
#p19FirstLast.txt
#(e.g. p19DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. 13, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 8

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

1. What is the explicit expression for the sequence
i. a(0) = 0, a(1) = 0, a(2) = 0, a(n) = 1 for n=3..infinity
x^3+x^4+...

ii. Use Maple to find an explicit expression for the generating function of 0,1,8,27,64,125,... a(n)=n^3
Sum(n^3*x^n,n=0..infinity)

2.
Find the egf of a(n) = n, for 0<=n<=5, a(n) = 0 for n>=6
x+2x^2+3x^3+4x^4+5x^5

3. What is the egf of a(n) = 0 for n=0..5 and a(n) = 1 for n>=6
Sum(x^n,n=6..infinity)

4. Find the egf of a(0) = 0, a(1) = 0, a(n) = (n-2)! for n>=2
Sum((n-2)!*x^n,n=2..infinity)

5.
i. What is the A number of this sequence?
The A number of the sequence is A001187.

ii. How many digits does the number of labeled connected graphs with 150 vertices have?
f := taylor(log(add(2^(((n - 1)*n)/2)*x^n/n!, n = 0 .. 152)), x = 0, 151);
[seq(i!*coeff(f, x, i), i = 1 .. 150)][150];
1023767986661322768993396667007831344315004313285307690628646458\

  016297531685367717317731059086210626[...3165 digits...]5760637\

  84642460313094749837660313430723573166372636046406007051172837\

  8754705040534724490510397865984

6. Use the technique of weight enumeration to find the exact number of sequences, a[1],a[2],...,a[r] (r can be any length) where each of the a[i] is a member of {3,4,7} that add up to 1001.
Not sure. How do we come up with an ogf for this?

7.
i. Is [{{1,3,4},{6,7}},52] a member of X(7)? Why?
Since X(n) is defined to have the form {[Permutation=pi,SetPartition=sp]}, [{{1,3,4},{6,7}},52] is not a member of X(7), but [52,{{1,3,4},{6,7}}] is.

8. How many triples of the form [Labeled Tree, Permutation, SetPartition] of size 150 (meaning that the number of vertices of the tree + length of permutation + size of set partition is 150)?
egf(X) = f = egf(Labeled Tree)*egf(Permutations)*egf(SetPartitions) = e^((e^x)-1)*(1/(1/x))*log(Sum(2^(((n - 1)*n)/2)*x^n/n!,n=0..infinity))

150!*coeff(taylor(log(add(2^(((n - 1)*n)/2)*x^n/n!, n = 0 .. 152))*exp(exp(x) - 1)/(1 - x), x = 0, 151), x, 150);
1023767986661322768993396667007831344315004743666659950655067397\

  462786173806045058426690027268911289[...3165 digits...]4218518\

  73674697708953438403937334134037869177800866625161779989780184\

  8391545445444461744791314757771