#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:

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

Q1. 
	(i). what is the explicit expression of GF a(n) = 0? 
	(ii). what is the explicit expression of GF a(n) = n^3? 

A1. 
	(i). 
		GF = a(0) * x^0 + a(1) * x^1 + ... 
	 		= 0 + 0 + 0 + ... = 0
	(ii). 
		GF = a(0) * x^0 + a(1) * x^1 + ... 
			= 0 + x + 8*x^2/2 + ... + 
			= [by maple] 
			= x(x^2 + 4x + 1) / (x-1)^4
	
Q2. 
	Find EGF of a(n) = n for 0<=n<=5 a(n) = 0 for n >= 6
A2.
	EGF = 0 * x^0 / 0! + ... 5 * x^5 / 5! + 0
		= 0 + x + 2x^2/2 + 3x^3 / 6 + 4x^4 / 24 + 5x^5 / 120
		= x+x^2+(1/2)*x^3+(1/6)*x^4+(1/24)*x^5

Q3.	
	What is the EGF of a(n) = 0 [n = 0..5] and a(n) = 1 [n >= 6]

A3. 
	Because EGF of a(n) = 1 is e^x. 
	So EGF of a(n) = 1 [n >= 6] = e^x - x - 2x^2/2 - 3x^3/6 - 4x^4/24 - 5x^5/120

Q4.
	Find the EGF of a(n) = 0 [n=0..1] and a(n) = (n - 2)! [n >= 2]

A4. 
	sum(factorial(n-2)*x^n/factorial(n), n = 2 .. infinity) = -x*ln(-x+1)+ln(-x+1)+x [maple]

Q5. 
	(i). What is the A-number of this sequence? 
	(ii). How many digits does the number of labeled connected graphs with 150 verticies have? 
	
A5.
	(i). A1187
	(ii). 
		f := taylor(log(add(2^((1/2)*n*(n-1))*x^n/factorial(n), n = 0 .. 151)), x = 0, 151);
		coeff(f, x, 150)*factorial(150);
			= a number with 3365 digits 

Q6.	
	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 one of the following set {3, 4, 7} that adds up to 1001 

A6. 
	c*x^1001 c = ? 

	|A|x = {O} + x^3|A|x + x^4|A|x + x^7|A|x

	|A|x = 1 + x^3|A|x + x^4|A|x + x^7|A|x 

	|A|x (1 - x^3 - x^4 - x^7) = 1

	|A|x = 1 / (1 - x^3 - x^4 - x^7)

	=========== MAPLE ==========

	f := 1/(-x^7-x^4-x^3+1)
	coeff(taylor(f, x, 1002), x, 1001) = 37327228629056835260060479719213438237141848693395064396004356099234232222305300761357555310103620202925186169




Q7. 
	Is [{{1,3,4},{6,7}}, 52] a member of X(7)? Why? 

A7.
	No, it is not because this is not a valid pair of permutation, setpartition.


Q8. labelled tree egf: 
	1 + sum(n^(n - 2)*x^n/n!,n=1..infinity)
	How many triples of the form
	[labeled tree, permutation, setpartition] of size 150 (meaning that the number of vertices of the tree + the length of the permutation + the size of the set that set partition partitions is 150) 
A8.
	egf(total) = egf(Labeled Trees) * egf(PERMUTATION) * egf(SetPartitions) = (1 + sum(n^(n - 2)*x^n/n!,n=1..infinity)) * exp(exp(x) - 1) * (1/(1 - x))

	evalf(coeff(taylor(1+(sum(n^(n-2)*x^n/factorial(n), n = 1 .. infinity))*exp(exp(x)-1)/(1-x), x, 151), x, 150)) = ~5.071792087*10^59

	NOT sure why i have a floating point number