#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

-----------------------------------
Attendance Question 1:
i) What is the explicit expression for the sequence a(0)=0, a(1)=0, a(2)=0, a(n)=1 for n=3,4,...

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

iii) 1,n,binomial(n,2),..., binomial(n,n)
Fix n, Let a(k) = binomial(n,k) = (n+k)! / (n!*k!)


i) x^3 * (1 + x + x^2 + ...)
The explicit expression is x^3/(1-x).


ii) 
assume(abs(x)<1);
sum(n^3*x^n, n = 0..infinity);
The explicit expression is x*(x^2+4*x+1)/(x-1)^4.


iii) (1+x)^n is the explicit expression.
-----------------------------------
Attendance Question 2:

Find the EGF of a(n)=n, for 0<=n<=5, a(n)=0 if n>=6.

0*x^0/0! + 1*x/1! + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! + 0 + ...
x + x^2 + x^3/2 + x^4/6 + x^5/24


-----------------------------------
Attendance Question 3:

What is the EGF of a(n)=0 for n=0,1,2,3,4,5 and a(n)=1 for n>=6?

exp(x) - 1 - x - x^2/2 - x^3/6 - x^4/24 - x^5/120


-----------------------------------
Attendance Question 4:

Find the EGF of a(0)=0, a(1)=0, a(n)=(n-2)! for n>=2

Sum(x^n * (n-2)!/n!, n = 2..infinity) = Sum(x^n/(n*(n-1)), n = 2..infinity)

-x*ln(-x+1)+ln(-x+1)+x

-----------------------------------
Attendance Question 5:

(i) What is the A-number of this sequence?

A1187.


(ii) How many digits does the number of labeled connected graphs with 150 vertices have?

f:=taylor(log(add(2^(n*(n-1)/2)*x^n/n!,n=0..151)),x=0,151):
length(150! * coeff(f,x,150))

The number of digits is 3365.

-----------------------------------
Attendance Question 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.

f := 1/(1 - x^3 - x^4 - x^7):
coeff(taylor(f,x=0,1002),x,1001);
37327228629056835260060479719213438237141848693395064396004356099234232222305300761357555310103620202925186169 is the number of sequences that add up to 1001 where each a[i] is a member of {3,4,7}.


-----------------------------------
Attendance Question 7:

(i) Is [{{1,3,4},{6,7}},52] a member of X(7)? Why?
No it is not a member of X(7) because we specifically defined X(n) to be a list where the first item is a permutation and the second item is a set partition so switching the places of the permutation and set partition no longer makes it valid.
 

-----------------------------------
Attendance Question 8:

Later on we will prove that the egf of labelled trees is 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 num of vertices of the tree + length of permutation + size of set that SetPartition partitions is 150)?

lt := 1+sum(n^(n-2)*x^n/n!, n=1..infinity):
pi := 1/(1-x):
sp := exp(exp(x)-1):
egf := lt * pi * sp:

coeff(taylor(egf,x=0,151), x, 150) * 150!;
28977095542425090391080657989766683580624301002611004688373441982498475144258790949730800839265420513593407126434947114816932428043259928872950627837484054712104869467896270548805752624407025846130055117327436957879166244375004554236854437226696641804705690475416777108324895809564401177362712104654213569067621944677124028
is the number of triples that matches the description.