#ATTENDANCE QUIZ FOR LECTURE 20 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p20
#with an attachment called
#p20FirstLast.txt
#(e.g. p19DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. 17, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 7

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

1. When was Rabbi Levi Ben Gerson born and when did he die?
Rabbi Levi Ben Gerson was born in 1288 and died in 1344.

2. When was E.T. Bell born and when did he die? What is he famous for? What is the name of the famous book about the history of mathematics that he wrote?
E.T. Bell was born in 1883 and died in 1960. He was famous his science fiction books as well as his writings on mathematics, and the name of his famous book about the history of mathematics that he wrote is Men of Mathematics.

3. What was the pen name of E.T. Bell?
The pen name of E.T. Bell was John Taine.

4. How many set partitions of a 300 element set are there with exactly 5 members?
Snk(300, 5);
 4090911221081438794246476628982333220442638220478190976605825883041810075846210707549539683575840119419320444839733557924380958558669263052623178486192842007420476226080784639399165079562132928175942227395000

5. Is this sequence in the OEIS? What is the A number if it is?
Sequence: [1,1,4,11,41,162,,715,3425,17722,98253,580317,3633280,24011157]
Yes, this sequence is in the OEIS. The A number is A000296.

6. Is this sequence in the OEIS? What is the A number if it is?
Sequence: [0,1,1,4,10,40,140,630,2800,14070,73150,412720]
No, this sequence is not in the OEIS.

7. Let a_k(n) be the number of permutations where every cycle is of length at least k
a_1(n)=n!, a_2(n)=d(n)
For k=3, k=4, k=5, ..., find the find the first 30 terms of a_k(n) and see whether they are already in the OEIS. Which k is the smallest that is not there?

#k=3:
egf3:= -log(1-x)-x-(x^2/2!)
f3 := taylor(exp(-log(1 - x) - x - x^2/2!), x = 0, 41);
seq(i!*coeff(f3, x, i), i = 1 .. 20);
 0, 0, 2, 6, 24, 160, 1140, 8988, 80864, 809856, 8907480, 106877320, 1389428832, 19452141696, 291781655984, 4668504894480, 79364592318720, 1428562679845888, 27142690734936864, 542853814536802656, 11399930109077490560, 250798462399300784640, 5768364635100620089152, 138440751242507472273856, 3461018781064593367693824, 89986488307675206245836800, 2429635184307185219369763200, 68029785160601345467104670848, 1972863769657440129000783404544, 59185913089723198139150966450176
#This is in the OEIS, the A number is A038205

#k=4:
egf4:= -log(1-x)-x-(x^2/2!)-(x^3/3!)
f4 := taylor(exp(-log(1 - x) - x - x^2/2! - x^3/3!), x = 0, 41);
seq(i!*coeff(f4, x, i), i = 1 .. 20);
 0, 0, 1, 6, 24, 130, 930, 7644, 68824, 685656, 7535340, 90457840, 1176171216, 16466275176, 246988107184, 3951794537040, 67180633256640, 1209252349298944, 22975793737934544, 459515836026817056, 9649832432640849280, 212296314644126875680, 4882815247362243598176, 117187565931600216644416, 2929689147731179453981824, 76171917838532634871190400, 2056641781659044077126158400, 57585969886697625971808647424, 1669993126714400894865685465344, 50099793801417192589876087970176
#This is not in the OEIS

#k=5:
egf5:= -log(1-x)-x-(x^2/2!)-(x^3/3!)-(x^4/4!)
f5 := taylor(exp(-log(1 - x) - x - x^2/2! - x^3/3! - x^4/4!), x = 0, 41);
seq(i!*coeff(f5, x, i), i = 1 .. 20);
 0, 0, 1, 5, 24, 130, 895, 7259, 65800, 658356, 7234215, 86772235, 1128043136, 15793597170, 236909199709, 3790544072665, 64439047006160, 1159901713071064, 22038133521682785, 440762728595139381, 9256017668688582280, 203632388424110113370, 4683544911007142001591, 112405077697364465285635, 2810126942480860743464824, 73063300516110209971967900, 1972709114035507833676795975, 55235855193051777120662297499, 1601839800591088168046023229520, 48055194017655158162354726374426
#This is not in the OEIS

#Thus the smallest k for which it is not in the OEIS is k=4.