#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

# Question 1: When was Rabbi Levi Ben Gerson's born? When did he die? 

# Answer: Born 1288 and died in 1344. 

# Question 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 he wrote?

# Answer: He was born on February 7, 1883. He died on December 21, 1960. He is famous for his book
# of biographical essays titled Men of Mathematics.

# Question 3: In addition to being a brilliant mathematician, E.T. Bell had a pen name. What was the name?

# Answer: His pen name is John Taine. 

# Question 4: How many set partitions of a 300-element set are there with exactly 5 members?

# Answer: Snk(300,5)= 4090911221081438794246476628982333220442638220478190976605825883041810075846210707549539683575840119419320444839733557924380958558669263052623178486192842007420476226080784639399165079562132928175942227395000

# Question 5: Is 1,1,4,11,41,162,715, 3425, 17722, 98253, 580317, 3633280, 24011157 in the OEIS?
# What is the A-Number? 

# Answer: Yes, it is A296.

# Question 6: Is 0,1,1,4,10,40,140,630, ... in the OEIS? What is the A-Number?

# Answer: No, it is not in the OEIS. 

# Question 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 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? 

# Answer: 

# The smallest k that is not there is k = 5. 

# a_3(n) egf -> -log(1-x)-x-x^2/2 = exp(-log(-1-x)) * exp(-x) * exp(-x^2/2)  = 1/(1-x)*exp(-x)*exp(-x^2/2) 

# f := exp(-x)*exp(-x^2/2)/(1 - x);
#                                     /  1  2\
#                          exp(-x) exp|- - x |
#                                     \  2   /
#                     f := -------------------
#                                 1 - x       

# [seq(coeff(taylor(f, x = 0, 31), x, i)*i!, i = 1 .. 30)];
# [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 A038205. 
# --------------------------------------------------

# a_4(n) egf: -log(1-x)-x-x^2/2 -x^3/3 = exp(-log(-1-x)) * exp(-x) * exp(-x^2/2) * exp(-x^3/3)  = 1/(1-x)*exp(-x)*exp(-x^2/2)*exp(-x^3/3)

# z := exp(-x)*exp(-x^2/2)*exp(-x^3/3)/(1 - x);
#                               /  1  2\    /  1  3\
#                    exp(-x) exp|- - x | exp|- - x |
#                               \  2   /    \  3   /
#               z := -------------------------------
#                                 1 - x             

# [seq(coeff(taylor(z, x = 0, 31), x, i)*i!, i = 1 .. 30)];
# [0, 0, 0, 6, 24, 120, 720, 6300, 58464, 586656, 6384960, 
#   76471560, 994831200, 13939507296, 209097854784, 3345235180560, 
#   56866395720960, 1023601917024000, 19448577603454464, 
#   388972171805410656, 8168409582839579520, 179704944537482689920, 
#   4133213636880538425600, 99197131945856677419456, 
#   2479928332674564111757824, 64478136551481363914841600, 
#   1740909683073851861225241600, 48745471108643784941792976000, 
#   1413618662349077210550915116544, 42408559873877916528445525618176]

# This is A047865. 
# --------------------------------------------------

# a_5(n) egf: -log(1-x)-x-x^2/2 -x^3/3 - x^4/4 = exp(-log(-1-x)) * exp(-x) * exp(-x^2/2) * exp(-x^3/3) * exp(-x^4/4)  = 1/(1-x)*exp(-x)*exp(-x^2/2)*exp(-x^3/3)*exp(-x^4/4)

# g := exp(-x)*exp(-x^2/2)*exp(-x^3/3)*exp(-x^4/4)/(1 - x);
#                         /  1  2\    /  1  3\    /  1  4\
#              exp(-x) exp|- - x | exp|- - x | exp|- - x |
#                         \  2   /    \  3   /    \  4   /
#         g := -------------------------------------------
#                                 1 - x                   

# [seq(coeff(taylor(g, x = 0, 31), x, i)*i!, i = 1 .. 30)];
# [0, 0, 0, 0, 24, 120, 720, 5040, 40320, 435456, 4959360, 
#   60255360, 782939520, 10870104960, 162642864384, 2602111599360, 
#   44265739714560, 797239080529920, 15148603957800960, 
#   302953165014675456, 6361636837067089920, 139952704431008286720, 
#   3218922790460015063040, 77254605493692206469120, 
#   1931370859803315548749824, 50215664438816457218457600, 
#   1355822305759073660955033600, 37963012534836747190338355200, 
#   1100927264115029271588399513600, 
#   33027818402324374018424155570176]

# This is not in the OEIS.