#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
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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..
with(gfun);
guessgf([0, 0, 0, 1, 1, 1, 1], x);
                         [    3       ]
                         [   x        ]
                         [- -----, ogf]
                         [  x - 1     ]

(ii) Use MAPLE to find an explicit expression for generating function of 0,1,8,27,64,125,... a(n)=n^3
guessgf([0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000], x);
              [          3      2               ]
              [        -x  - 4 x  - x           ]
              [- --------------------------, ogf]
              [   4      3      2               ]
              [  x  - 4 x  + 6 x  - 4 x + 1     ]


(iii) 1,n,binomial(n,2), ..., binomial(n,n). Fix n, let a(k)=binomial(n,k) = (n+k)!/(n!*k!)
a := k -> (n + k)!/(n!*k!);
a := proc (k) options operator, arrow; factorial(n+k)/(factorial\

  (n)*factorial(k)) end proc


t := [seq(a(i), i = 0 .. 10)];
     [   factorial(n + 1)  factorial(n + 2)  factorial(n + 3)  
t := [1, ----------------, ----------------, ----------------, 
     [     factorial(n)     2 factorial(n)    6 factorial(n)   

  factorial(n + 4)  factorial(n + 5)  factorial(n + 6)  
  ----------------, ----------------, ----------------, 
  24 factorial(n)   120 factorial(n)  720 factorial(n)  

  factorial(n + 7)    factorial(n + 8)    factorial(n + 9)    
  -----------------, ------------------, -------------------, 
  5040 factorial(n)  40320 factorial(n)  362880 factorial(n)  

   factorial(n + 10)  ]
  --------------------]
  3628800 factorial(n)]

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2. Find the EGF of a(n) = n, for 0<=n<=5, a(n)=0 if n>=6
A2. EGF is := a(0)*x/0! + a(1)*x/1! + a(2)*x/2! + a(3)*x/3! + a(4)*x/4! + a(5)*x/5! + 0
let a(0)=0, a(1)=4, a(2)=-4, a(3)=6, a(4)=-6, a(5)=24
1 + 4x + (-2)x + x + (-1/4)x + (1/5)x = 1 + 3x + (1/20)x
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3. What is the EGF of a(n)=0 for n=0,1,2,3,4,5 and a(n)=1 for n>=6
A3. guessgf([0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], x);
                         [    6       ]
                         [   x        ]
                         [- -----, ogf]
                         [  x - 1     ]

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4. Find the EGF of a(0)=0, a(1)=0, a(n)=(n-2)! for n>=2
A4. a(0) := 0;
a(1) := 0;
                           a(0) := 0

                           a(1) := 0

seq((n - 2)!, n = 2 .. 10);
             1, 1, 2, 6, 24, 120, 720, 5040, 40320

guessgf([0, 0, 1, 1, 2, 6, 24, 120, 720, 5040, 40320], x);
              [-ln(1 - x) x + ln(1 - x) + x, egf]

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5. (i) What is the A-number of the sequence [seq(i!*coeff(f,x,i),i=1..15)];
	A-number := A001187
(ii) How many digits does the number of labelled connected graphs with 150 vertices has?
evalf(2^((150*(150 - 1))/2));
                                     3364
                       1.023767987 10    
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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
A6.  
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7. Is [{{1,3,4},{6,7}},52] a member of X(7) ? Why?
A7. yes it is a member. 
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8. How many triples of the form [labelled trees, 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 
SetPartitions is 150)
A8. 150^(150 - 2);
1152142879772419975975490874746881132725649004872810390595996610\

  08670835565381399514928799097290248121972347799892552888829796\

  96643430247982031744413689011707901954650878906250000000000000\

  00000000000000000000000000000000000000000000000000000000000000\

  00000000000000000000000000000000000000000000000000000000000000\

  00000000000

nops(permute(150));
57133839564458545904789328652610540031895535786011264182548375833179829124845398393126574488675311145377107878746854204162666250198684504466355949195922066574942592095735778929325357290444962472405416790722118445437122269675520000000000000000000000000000000000000*with(combinat, bell);
                             [bell]

bell(150);
6820641270431347230881435684901405911110305976711784053477825671\

  87230042265005133702324247993318753907221842250901468725996068\

  62902821231486434088755904062876982601314838951289688988786630\

  77403


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