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How is binomial(n, n+p) computed when p ...

sand15 765 Maple
binomial gamma
February 29 2024
1 1

While answering a question on this site I accidentally met expressions of the form binomial(n, min(n, r)+1) where both n and r are positive integers and n is strictly lower than r.

For the record the common definition of the binomial coefficient binomial(n, k) is based on the double inequality 0 <= k <= n  and the only generalized definition where k could be larger than n I know of is the NegativeBinomial distribution where we use
binomial(-n, k) which, with 0 <= k <= n  again makes the first operator lower than the second.

I tried to understand how Maple does this

binomial(n, min(n, r)+1) assuming n < r,  n::posint
                               0

(more generallyn, for any strictly positive integer p, binomial(n, min(n, r)+p) = 0 under the assumptions above)

I guess that the explanationrelies upon what I did to get the output (2) in the attached file.
Can you confirm/infirm this and, as I wasn't capable to find any clue in help(binomial), [Maple 2015], if the way maple computes
these results is documented elsewhere.

Thanks in advence.

> 

restart:

 

Let us start with this result

 

> 

t0 := binomial(n, min(n, r)+1);
eval(t0) assuming n < r;
eval(%) assuming n::posint;

# I didn't find in help(binomial) the argument used to get this last result.

binomial(n, min(n, r)+1)

  

binomial(n, n+1)

  

0

 (1)
> 

# What happens if binomial is converted into factorials

t1 := convert(t0, factorial);
eval(t1) assuming n < r;

factorial(n)/(factorial(min(n, r)+1)*factorial(n-min(n, r)-1))

  

Error, (in assuming) when calling '`one of {eval, min, factorial}`'. Received: 'numeric exception: division by zero'

  
> 

# Or into GAMMA function?

t2 := convert(t1, GAMMA);
eval(t2) assuming n < r;

GAMMA(n+1)/(GAMMA(min(n, r)+2)*GAMMA(n-min(n, r)))

  

Error, (in assuming) when calling 'GAMMA'. Received: 'numeric exception: division by zero'

  
> 

# Try to replace min(n, r) = n by n-epsilon and take the limit as epsilon goes to 0
# from the right.

t3 := algsubs(min(n, r) = n-epsilon, t2);
limit(t3, epsilon=0, right)

GAMMA(n+1)/(GAMMA(n-epsilon+2)*GAMMA(epsilon))

  

0

 (2)

 

We recover here the result (1), but does Maple really proceed this way?

Download binomial.mw

Why is the output of printf delayed?...

sand15 765 Maple 2015
printf
December 09 2023
0 5

When there are print commands in a loop their content is printed as soon as this command is executed.
This is not the case with printf whose displays are delayed (buffered?).
Is there a way to force the display of printf when the command is executed?

TIA

Motivation: I want to display intermediate execution times in a prettier way than print offers.

Can we increase the presision of the sli...

sand15 765 Maple 2015
explore precision slider
December 09 2023
2 6

Is it possible to enlarge the sliders in Explore(plot(...), ...) and increase their "resolution" (meaning to have a higher precision when the slider is moved)?
If Maple does offer this option, could you tell me from what version this is the case

TIA

How can we prove this equality?...

sand15 765 Maple
simplification equality
November 16 2023
4 2

I'm stucked in trying to prove that rel(n)  is true for each integer n > 1.

> 

restart
> 

rel := n -> (n-3)^(n/(n-1))*2^(n/(n-1))-((n-1)*2^(n/(n-1))-4*2^(1/(n-1)))*(n-3)^(1/(n-1)) = 0

proc (n) options operator, arrow; (n-3)^(n/(n-1))*2^(n/(n-1))-((n-1)*2^(n/(n-1))-4*2^(1/(n-1)))*(n-3)^(1/(n-1)) = 0 end proc

 (1)
> 

 

Download Prove_It_True.mw

Do you have any idea to do this?

TIA

Problems while trying to upload a mw fil...

sand15 765 Maple , MaplePrimes
upload
October 31 2023
0 2


For several hours I experience difficulties to upload a mw file (file LaneEmdenNumeric.mw to answer this question).

Given the error message I get

I thought it was a general technical problem.

But then I realized that this file ColumnSorting_4.mw , for instance, could be uploaded.

Nevertheless, as soon as I try to upload LaneEmdenNumeric.mw I get the error displayed above and it becomes impossible to upload any other file, even ColumnSorting_4.mw.

Does someone has any idea about what's going on?
TIA

For the record, LaneEmdenNumeric.mw contains nothing special that could prevent its uploading.
This file can be uploaded here  DropBox  (replace the txt extension by mw before opening it with Maple).

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