#ATTENDANCE QUIZ FOR LECTURE 14 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p24
#with an attachment called
#p24FirstLast.txt
#(e.g. p24DoronZeilberger.txt)

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


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 10

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

1) Write down manually the set / integer partitions of 7

1+1+1+1+1+1+1,
1+1+1+1+1+2,
1+1+1+1+3,1+1+1+2+2
1+1+1+4,1+1+3+2,1+2+2+2
1+1+5,1+2+4,1+3+3,2+2+3 
1+6,3+4,2+5
7

2) Write down manually the set of partitions of 7 into distinct parts, write the number of integer partitions into odd parts. Find out if they are the same number.

Distinct Parts: 5
1+2+4,
1+6,
3+4,
2+5,
7

Odd Parts: 5
1+1+1+1+1+1+1,
1+1+1+1+3,
1+1+5,
1+3+3,
7

Yep, they are the same number.

3) What is the A-number of this sequence?
1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490
A000041

4) Who wrote the classic book on the theory of partitions? What is his birthdate?
George Andrews, December 4, 1938. He is still alive.

5) What is the A-number of this sequence?
1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,38,46,54,64
A000009

6) What is the a-number of this?
1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23,26,31
A003114

7) Whose theorem is it that these above two sequences are the same?
referring to:
seq(nops(PnD(i,2)),i=1..20)
seq(nops(PnC(i,{1,4},5)),i=1..20); 
It is the Rogers-Ramanujan identity. Rogers first discovered it and then Ramanujan rediscovered it.

8) Is it in the OEIS? What is the A-number of this sequence?
6,27,98,315,913,2462,6237,15035...
Yes, A071734

9)
(i) Does there exist a k between 0 and 6 such that seq(pn(7*n+k)/7,n=1..60); is an integer sequence?
What is the A-number?
Yes, k=5
11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287, 7098041203, 13951818365, 27047831797, 51760979985, 97851055848, 182858720324, 338003248835, 618337665521, 1120093746591, 2010077957041, 3575124822873, 6304704983465, 11027984666216, 19139751334984, 32970507766383, 56389096665051, 95778303294310, 161605500562658, 270937729084512, 451448266735571, 747767295679096, 1231499467923575, 2016951809325715, 3285715236498191, 5324915110964011, 8586478548523792, 13778587170314680, 22006228197082290, 34986362493626058, 55376076131289889, 87271200535983443, 136961242558905435, 214068391941511583, 333260171220556048, 516816109123943489, 798461886085057784, 1229078822764093580, 1885179852417919386, 2881454716972276237, 4389283592442150157

A071746

(ii) Does there exist a k between 0 and 10 such that seq(pn(11*n+k)/11,n=1..60); is an integer sequence?
What is the A-number?
Yes, k=6
27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805845, 1117485621, 3366123200, 9767105406, 27398618368, 74534264393, 197147918679, 508189847045, 1279140518117, 3149375120229, 7596463993261, 17975137880152, 41776886404425, 95472499084647, 214747195030701, 475851915432152, 1039594734347464, 2240918740984590, 4769276748075331, 10027987311390252, 20842935170110650, 42846733115308980, 87157154355667095, 175514188395496892, 350048948606129926, 691709349394534479, 1354741480937268088, 2630730659734981650, 5066678649512389696, 9681186536839919871, 18357579935010426239, 34554133523920054407, 64579072566966203455, 119865491055660930293, 221006373119026120716, 404870518142812813398, 737082082866134011286, 1333788661167343930225, 2399432817727762088014, 4291959601287308912758, 7634832928240080000211, 13508538678919990017965, 23776449251021876569610, 41636798909831300777602, 72553524269818107818797, 125819244166493359868989, 217168633063630216612362, 373130181055092154587010, 638243703049886314995807, 1086984023480585956148135

A076394

10) Why Should Dr. Z. From now on make you watch the lectures before the recitation?

Watching videos before recitation is good because if we have any burning questions from the video, we can ask you at the recitation. (I usually watch the videos before recitation but I wait until after recitation to do the participation quiz).