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#2151 2013-02-16 04:18:10

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

What table?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#2152 2013-02-16 04:24:59

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

1;
1, 1, 1;
1, 2, 4, 6, 6;
1, 3, 9, 24, 54, 90, 90;
1, 4, 16, 60, 204, 600, 1440, 2520, 2520;
1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400;
1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400,
It is in the examples in the link I posted.

The line
1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400;
is the one of interest.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2153 2013-02-16 04:30:41

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Hmmm, how do match that up with the pdf? Please demonstrate.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2154 2013-02-16 04:34:38

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

Hi bobbym

Do you see his table for P(X=n) on page 18?

Let T_n be the nth number in the 6th line of the triangle
1;
1, 1, 1;
1, 2, 4, 6, 6;
1, 3, 9, 24, 54, 90, 90;
1, 4, 16, 60, 204, 600, 1440, 2520, 2520;
1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400;
1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400,
.
.
.
Then P(X=n)=T_(n-3)*Binomial[n-1,2]/6^(n-1)

Last edited by anonimnystefy (2013-02-16 05:10:07)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2155 2013-02-16 04:36:32

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Then what is i?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#2156 2013-02-16 04:38:28

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

Sorry, wrong variable. Check again.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2157 2013-02-16 04:39:56

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Okay, you were supposed to check that but maybe you are not at your computer. I can check it. Supposing you are right is there an easier way to generate those numbers than he did?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#2158 2013-02-16 04:42:02

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

I checked the for values of n from 3 to 9. I did not check the last 4.

I think it might be. I do not understand yet how the triangle was generated. I will have a look later.

Edit: OEIS gives a formula for those: T(k,n) = n![x^n](1+x+x^2/2)^k.

Last edited by anonimnystefy (2013-02-16 04:43:26)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2159 2013-02-16 04:44:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Okay, I will need to work on it.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2160 2013-02-16 04:50:38

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

For this particular problem the formula is T(n) = n![x^n](1+x+x^2/2)^5, which is kinda obvious, considering the fact that it is the number of ways we can put n marbles into 5 jars, with both jars being different and order of putting in mattering.

I think we actually solved it!

Last edited by anonimnystefy (2013-02-16 04:50:56)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2161 2013-02-16 04:52:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Hold on! You solved it, but so far it is not working.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2162 2013-02-16 04:55:53

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

Why not?

With the following code I get the correct answer!

sum(i!/(2*6^(i-1))*coeff(ratexpand((1+x+x^2/2)^5),x^(i-3)),i,3,13),numer;

Last edited by anonimnystefy (2013-02-16 05:42:29)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2163 2013-02-16 04:57:25

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

I was talking about the T[n] idea.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2164 2013-02-16 05:02:29

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

It is working for me. What's the problem?

I think it could be generalized for 4, 5 and more dice in a row.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2165 2013-02-16 05:06:41

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Calm down please.

You are getting this to work?

Last edited by bobbym (2013-02-16 05:06:58)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2166 2013-02-16 05:09:17

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

I am calm. Sorry if I sounded frustrated.

T(n-3), not T(n), my bad. Edited the original post, too.

Last edited by anonimnystefy (2013-02-16 05:10:24)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2167 2013-02-16 05:13:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

I am kidding you. Just being funny.

I have it working now. Okay, you say you can improve on that formula?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2168 2013-02-16 05:34:24

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

I think it is plausible. Give me 5 minutes.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2169 2013-02-16 05:35:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

I am working on it also.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2170 2013-02-16 05:37:22

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

Hi bobbym

I am getting 11.21375706013234, or

in closed form.

Edit: Here's my code:

sum(i*(i-4)!/(6^(i-1))*binomial(i-1,3)*coeff(ratexpand((1+x+x^2/2+x^3/6)^5),x^(i-4)),i,4,19);

Last edited by anonimnystefy (2013-02-16 05:40:07)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2171 2013-02-16 05:40:12

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

You are jumping ahead, what is that the answer to?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2172 2013-02-16 05:41:32

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

Expected number of throws till we get a number 4 times.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2173 2013-02-16 05:43:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Please hold on with that. We are still working on the 3. I agree that it will be possible to  generalize but first I would like to simplify how you are doing the three. If we can do that we can really present it to him as a superior solution.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#2174 2013-02-16 05:45:32

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,836

Re: What do you think?

Ok. So what do we do now?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#2175 2013-02-16 05:47:26

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,601

Re: What do you think?

Is there a way to generate the sequence T[5,n] that does not require us to expand a trinomial or a sum with a ceiling function in it?

Last edited by bobbym (2013-02-16 05:47:45)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

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