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Arrive five minutes early (but start on time)
Write on board: http://www.math.wisc.edu/~propp/491
Announce that not all the information was correct: in particular,
the reading list has been changed, and the office hours are
still being worked out.
How do you feel about Steep and Brew?
On-line lecture notes: the meaning of УЕФ
Classroom culture. Gender.
Go around and say who you are, why youТre here and what you expect to get out of the course, and what movie you liked best this summer
Office hours: to be arranged (partly based on your schedules).
Have students fill out cards about themselves (give them 3-4 minutes):
Name (official name, plus what you like to be called,
CIRCLED)
School, year
Interests (academic and non-)
Major (if declared)
Dorm or approximate address
Phone number
Email address
Scheduling constraints (for SSL and office hours)
Anything else I should know
Explain why I do this (if you feel uncomfortable about including
this information, let me know)
Phone list will be available over email tomorrow
You are encouraged to form study groups.
УDo not abuse this!Ф
If someone calls you with a question, try to be helpful.
If you have trouble finding a study partner, see me.
Until I set up regular office hours, feel free to ask me questions by
email, or to try to set up an appointment.
If youТre stuck on a problem, it might be a good idea to check you email for hints or corrections.
Have the students hand in the cards AND pick up the handouts
Assignment #1 (due in class, or by email before class, next Thursday, not next Tuesday)
Demo: The sum of the squares of the elements in the nth row? ...
1, 2, 6, 20, 70, 252, ...
Maple: If you type Уsum(binomial(5,k)^ 2,k=0..5);Ф after the
prompt, Maple says У252Ф.
Or: Уf := proc(n) RETURN(sum(binomial(n,k)^2,k=0..n)); end;Ф
and Уf(5);Ф
What about Уf(0)Ф? What do you think is going on?
Е Try binomial(0,k);
Moral: DonТt blindly trust a computer!
When you find things like this, please send me email; weТll
compile a list of snares for the unwary.
How can we fix this?
f := proc(n) if n=0 then 1 else sum(binomial(n,k)^2,k=0..n); fi;
end;
(Try g(5).)
Pattern hunting: ratios!
2/1 = 2, 6/2 = 3, 20/6 = 10/3, 70/20 = 7/2, 252/70 = 18/5, ...
seq(f(n+1)/f(n),n=0..12);
LetТs rewrite 7/2 as 14/4:
2/1, 6/2, 10/3, 14/4, 18/5, ...
Aha!
seq((n+1)*f(n+1)/f(n),n=0..12);
seq((n+1)*f(n+1)/f(n)-4*n,n=0..12);
f(5) = (f(5)/f(4)) (f(4)/f(3)) (f(3)/f(2)) (f(2)/f(1))
= (18/5)(14/4)(10/3)(6/2)(2/1)
= 2^5 (9)(7)(5)(3)(1)/(5)(4)(3)(2)(1)
(9)(7)(5)(3)(1) = 9!/(8)(6)(4)(2) = 10!/(10)(8)(6)(4)(2)
= 10! / 2^5 (5)(4)(3)(2)(1) = 10! / 2^5 5!
f(5) = 10! / 5! 5!
So we conjecture:
sum_{k=0 to n} {n choose k}^2 = {2n choose n}.
Handbook of Integer Sequences (mention link to Sloane from the
SSL home-page)
www.research.att.com/~njas/sequences/
http://www.ams.org/notices/200308/comm-sloane.pdf
[What about
(1)(1) = 1,
(1)(2)+(2)(1) = 4,
(1)(3)+(3)(3)+(3)(1) = 15,
(1)(4)+(4)(6)+(6)(4)+(4)(1) = 56,
(1)(5)+(5)(10)+(10)(10)+(5)(5)+(5)(1) = 210, Е?
Conjecture answer together.
Algebraic proof?
Combinatorial proof?]
Get them doing Maple. Start them doing the homework! (But be sure to tell them that we wonТt ordinarily spend class time in this way. Encourage them to work in groups. Remind them to give credit to collaborators, and to record how much time they spend.)
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