Getting the coefficient.

anonimnystefy · 2013-12-17 07:55:38

Hi bobbym

I am getting

for the coefficient of ...

Last edited by anonimnystefy (2013-12-17 07:55:51)

bobbym · 2013-12-17 07:57:16

What are you getting for the coefficient of x^50000?

anonimnystefy · 2013-12-17 08:30:15

Sorry, that is the coefficient of 50000.

bobbym · 2013-12-17 08:42:50

That is not what I am getting. See you in a bit.

anonimnystefy · 2013-12-17 08:56:52

My recurrence is:

Last edited by anonimnystefy (2013-12-17 10:36:16)

bobbym · 2013-12-17 10:26:47

The recurrence should only be for the numerator.

anonimnystefy · 2013-12-17 10:30:16

It does not matter. The answer will be the same.

bobbym · 2013-12-17 10:31:09

I suggest you try it first before saying that.

anonimnystefy · 2013-12-17 10:45:04

The results are the same.

bobbym · 2013-12-17 10:49:41

Could be because your recurrence is wrong.

Use this one:

RecurrenceTable[{a[n] == 
     1/(-10000 + 
       n) (60005 a[-5 + n] - n a[-5 + n] + 50004 a[-4 + n] - 
        n a[-4 + n] + 40003 a[-3 + n] - n a[-3 + n] + 
        30002 a[-2 + n] - n a[-2 + n] + 20001 a[-1 + n] - 
        n a[-1 + n]), a[10000] == 1, a[10001] == 10000, 
    a[10002] == 50005000, a[10003] == 166716670000, 
    a[10004] == 416916712502500}, a, {n, 10000, 50000}] // Last;

anonimnystefy · 2013-12-17 10:53:33

I know exactly what's wrong.

bobbym · 2013-12-17 10:55:01

What is wrong?

anonimnystefy · 2013-12-17 11:00:34

I am using the wrong GF!

bobbym · 2013-12-17 11:04:22

Well, that will certainly cause a few problems. What will you do about that?

As soon as you get the correct answer you will see that there is an even bigger problem looming on the horizon. Also, then will come Part 3.

ElainaVW · 2013-12-20 06:20:13

Hello

I see the problem! When will you post part 3?

bobbym · 2013-12-20 07:05:24

Part 3

Our problem is that the 2 methods provided agree for small problems but disagree on the large one. Why is this?

Before we can answer that we need to find out which answer is correct so we do the problem in a third way.

With 10000 throws of a die what is the coefficient of x^50000?

This is equivalent to the diophantine equation:

a1 + a2 + a3 + ... a10000 = 50000

with 1 <= a1,a2,...a10000 <= 6

We make a change of variable.

a1 = a1+1, a2 = a2+1, a3 = a3+1, ...a10000=a10000+1

we get

a1 + a2 + a3 + ... + a10000 +10000 = 50000

this becomes

The equation becomes

a1 + a2 + a3 + ... + a10000 = 40000

0 <= a1,a2,...,a10000 <= 5

with the answer being the coefficient of x^40000

The gf for that is:

You should recognize that

are the coefficients with no restrictions.

Now

Only the terms x^6 to x^39996 could possibly contribute to the coefficient of x^40000, making the final answer:

The whole process looks a lot like a PIE so it can be generalized to the following series. For,

with

In Mathematica speak:

r = 40000;
n = 10000;
m = 5;

ans = Binomial[r + n - 1, n - 1] + 
   Sum[(-1)^
     k Binomial[n, k] Binomial[(r + n - 1) - (m + 1) k, n - 1], {k, 1,
      Floor[r/(m + 1)]}];

To get the probability of the sum being 50000:

Which agrees with Part 2's answer. We can now safely say that the result in Part 1 is incorrect.

Okay, what went wrong?

The nice guy wrote:

Numerical methods are not infallible and some sort of round off error is destroying the precision of the complex analysis method. This happens and we should forgive those mathematicians who love it so much.

bobbym ( the mean guy) wrote:

That method is typical of guys who think their beloved complex analysis is truly amazing and naively think what they know from textbooks works in the real world. Kaboobly doo!

Math Is Fun Forum

#26 2013-12-17 07:55:38

Re: Getting the coefficient.

#27 2013-12-17 07:57:16

Re: Getting the coefficient.

#28 2013-12-17 08:30:15

Re: Getting the coefficient.

#29 2013-12-17 08:42:50

Re: Getting the coefficient.

#30 2013-12-17 08:56:52

Re: Getting the coefficient.

#31 2013-12-17 10:26:47

Re: Getting the coefficient.

#32 2013-12-17 10:30:16

Re: Getting the coefficient.

#33 2013-12-17 10:31:09

Re: Getting the coefficient.

#34 2013-12-17 10:45:04

Re: Getting the coefficient.

#35 2013-12-17 10:49:41

Re: Getting the coefficient.

#36 2013-12-17 10:53:33

Re: Getting the coefficient.

#37 2013-12-17 10:55:01

Re: Getting the coefficient.

#38 2013-12-17 11:00:34

Re: Getting the coefficient.

#39 2013-12-17 11:04:22

Re: Getting the coefficient.

#40 2013-12-20 06:20:13

Re: Getting the coefficient.

#41 2013-12-20 07:05:24

Re: Getting the coefficient.

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