#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 

#1. 
#i. What is the explicit exp. for the sequence
#a(0) = 0; a(1)=0; a(2)=0; a(n)=1 for n=3,4,...

#answer: a(n) = n(n-1)(n-2)/n!

#ii. use maple to find an explicit exp for the generating function of a(n)=n^3

#answer: generating function is x*(x^2 + 4*x + 1)/(1 - x)^4

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

#answer: f(x) = 1*x/0! + 1*x/1! + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! +0
f(x) =  1+x+x^2+x^3/2+x^4/6+x^5/12

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#3. EGF of a(n) = 0 for [0,5] & a(n)=1 for n>=6

#answer: f(x) = x^6/6!+ ..... = exp(x) - (1+x+x^2/2!+x^3/3!+x^4/4!+x^5/5!)

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

#answer: f(x) = sum(x^n*(n-2)!/n!, n=0..infinity) - x = sum(x^n/(n(n-1)),n=0..infinity) - x

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#5. i)what is the A-number of this sequence.

f:=taylor(log(add(2^(n*(n-1)/2)*n^n/n!,n=0..101)),x=0,101);
[seq(i!*coeff(f,x,i)i=1..15)];

[1, 1, 4, 38, 728, 26704, 1866256, 251548592, 66296291072, 34496488594816, 35641657548953344, 73354596206766622208, 301272202649664088951808, 2471648811030443735290891264, 40527680937730480234609755344896]

#A001187, Number of connected labeled graphs with n nodes.


#ii) how many digits does the number of labeled connected graphs with 150 vertices has?

f := taylor(log(add(2^(n*(n - 1)/2)*x^n/n!, n = 0 .. 151)), x = 0, 151);
length(i!*coeff(f, x, i), i = 150);
                             180892
# 180892
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#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

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#7. i) is this [{1,3,4,},{6,7},52] a member of x(7)

#answer: no. The form is not right in the first place

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#8. How many triples of the form [labeled tree, permutation, setpartition] of size 150 ?

#egf of f = egf(labeled tree)*egf(perm)*egf(setpart) = (1+sum(n^(n-2)*x^n/n!,n=1..infinity))*exp(exp(x)-1)/1-x

coeff(taylor((1+sum(n^(n-2)*x^n/n!,n=1..infinity))*exp(exp(x)-1)/1-x,x=0,151),x,150)*150!;
18135887626188509526685748386228194637219765831967839453246596447444919901655563080888953096123452025563387299833597385857524965371627070270139497030545073124327808640547675061584108124067488576665711461385847073068067571813786819513432404168342460292146822434210479682262060824948101586419994892187324039077148160323196678