#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

# 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!)  

# Answer:
# (i) f(x) = x^3 + x^4 + ....  The explicit expression of f(x) = x^3/(1-x) 
# (ii) sum(x^n*n^3, n = 0 .. infinity) = x*(x^2 + 4*x + 1)/(x - 1)^4
# (iii) sum(binomial(n, k), k = 0 .. n) = 2^n 

# Question 2: 

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

# Answer: 
# x + 2x^2/2! + 3x^3/3! + 4x^4/4! + 5x^5/5! = x + x^2 + x^3/2 + x^4/6 + x^5/24

# 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

# Answer: 
# x^6/6! + x^7/7! + x^8/8! + .... = exp(x) - 1 - x - x^2/2! - x^3/3! - x^4/4! - x^5/5! =
# Sum(x^n/n!, n = 6..infinity) 

# Question 4

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

# Answer:

# x^2/2! + x^3/3! + 2! * x^4/4! + 3! * x^5/5! + 4! * x^6/6! + .... = 
# Sum((n-2)!/n! * x^n, n = 0..infinity) = Sum(x^n/(n*(n-1)), n = 0 .. infinity) 

# Question 5

# (i) What is the A-Number of this sequence ([1,1,4,38,728, 26704....])?
# (ii) How many digits does the number of labeled graphs with 150 vertices have?

# Answer:

# (i) The A-number is A1187. 

# (ii) f:=taylor(log(add(2^(n*(n-1)/2)*x^n/n!, n=0..200)),x=0,200):
# 150! * coeff(f,x,150) = 
# 10237679866613227689933966670078313443150043132853076906286464580162975316853677173177310590862106268927966079623397135234574364986806971998505744804516420114879839684936331772044141241976248133100789617901466906076056647995584686898165579874224400195550418621657599887307182534736311702295520309280470173143063202535320197918793788123330881271595669984163071912764545723178727014206344921648094040027007522283290304275430056814954257311428281943549877772625609033175388791853784210081841124194091541870448704782938393756397594653304611738777465484934381234280952644376525251074218052135732602315906061254822922271572666050503251028778145484484755315029010674746858581260308572883178732304310594971118010798026163487552029999617180796489870244320155510769701611740565079958379119412966382133456516914858242363228378390818237171956637083229694379044094480925855977341670365164213336480396037969636545560624089084106037496142833180644995003162240308231866989382501493268553727195439019455786059125396803763911243779855810193# 91207998062403869504920808442344866603107447567480750091216737300532350509890480154717100382634136771878562338909150653710386794336328634011498764389751476912502328788420243385525169115769992330838909846519260068387063083480970403591687038843867049497596203549765962566530241480775792683960469522357260545748561423065997841556065708339015859700399720858271599349224254960446774080247489470213965340460826844025235996524012705900736047043085524718927756796191178178404964315581471372318163856578360107267306324914499707960785721447904427280483509098300098097564859355120660484250150030056144660382687294839752466576867662073432461261267547405023565510468179172862728437397024707467813752077344187552882919167335077115548877972293192381870289288831903684652835175075560942427220361687610988089272833764038765652002722324030013755386287488326922966334421236566086036817850554766259632272112441047299413741894307109554642168788165022866393491559494467454529759510847050963593677574669593281811858661444025338237675371759631335# 94914832873616636349548070956635221475137096166430799097958612722264568361108374462133997187224300727321046894923651403723243191513407253279458698211583699246041152856258468677629784674280064779555823993088170553168851526843579476366093331346413867503272575454662653821642276927854413442391115646702641118275454622512817449245272605221962397591760679611845187550454546087060285348259513659218440398732942671821137867887461449077896305010849201989985656989511877896917428203269263255504772778571095434158747900142090162401995333226513643220168591756369005220598027492744979613422843058457780420627445675109390808216480708231732560689286977600958951475564588485788048514366602318993636874607496598797546724074624887717377551522406707874370343974156311726865786834307720680145202983030341861690637746970509787472563644102901741154960244599936066821846321442442386584734677609420374809852201658955659245700666724942559570206846648014476162436431697472411774514015593930534920203786613814959997554709058018249485657373905608713# 50732829784978101763257536871142822444833681032414299805024754408512460500035195493404280541184999668777276994027282683231187539152169794835486485949015680561179515381627213593155657819615310469950515760637846424603130947498376603134307235731663726360464060070511728378754705040534724490510397865984

# The number of digits of the above number is 3070.

# 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.

# Answer: 

# h := taylor(1/(-x^7 - x^4 - x^3 + 1), x = 0, 1002);
# coeff(h, x, 1001) = 
# 37327228629056835260060479719213438237141848693395064396004356099234232222305300761357555310103620202925186169 sequences

# Question 7
# (i) Is [{{1,3,4},{6,7}}, 52] a member of X(7)? Why? 
# (ii) How big is X(100)?

# Answer: 

# (i) [{{1,3,4},{6,7}}, 52] is a member of X(7) because the length of the permutation 52 is 2, the size of the 
# set that SP is partitioning is 5, and 5+2=7. The labels are all distinct as well as there are no repeats.

# (ii) The size of X(100) is:
# coeff(taylor(exp(exp(x)-1)/(1-x),x=0,101),x,100)*100! = 520288193827224460161665693343439067946826765691270139531070491212113377307944087322830833805396364699434666974424650987403343482352896062075174445266247411051

# 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 that the number of 
# vertices of the tree + the length of the permutation  + the size of the set that SetPartition partitions is exactly 150) 

# Answer:

# coeff(taylor(exp(exp(x) - 1)/(1 - x) + sum(n^(n - 2)*x^n/n!, n = 1 .. infinity), x = 0, 151), x, 150)*150! = 
# 11521428797724199759754908747468811327256490048728103905960284618681740482591778655985
# 955690495096915346505062068574473194663951321057127308414735985486825065039699403374409380177855791232
# 293320946953349084003636144732439152743129626016451562712564579589029606563571096685960973346192150974495
# 141451920139858528771934444653