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• I believe that I committed a mistake on having compiled (I use MATHEMATICA) the functions of the second identity. I am afraid that this second identity is not
Message 1 of 4 , Dec 31, 2006
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I believe that I committed a mistake on having
compiled (I use MATHEMATICA) the functions of the
second identity.

I am afraid that this second identity is not true.

I wait for your commentaries on the first and third
identity.

Sincerely

Sebastián Martin Ruiz

--- David Cleaver <wraithx@...> escribió:

>
>
> Note:
> LHS stands for Left Hand Side
> RHS stands for Right Hand Side
>
> Sebastian Martin wrote:
> >
> >
> > Dear colleagues:
> >
> > I have found some identities:
> >
> > 1) Product[k^Floor[n/k],{k,1,n}]=
> > Product[Floor[n/k]!,{k,1,n}]
>
> I was able to confirm this for n=5 and n=6.
>
> >
> > 2) Product[Floor[n/k]^k,{k,1,n}]=
> > Product[k!,{k,1,n}]
>
> I was unable to confirm this, ie:
>
> For n = 5
> LHS = 20
> RHS = 34560
>
> For n = 6
> LHS = 432
> RHS = 24883200
>
> Did you perhaps type the wrong formula in?
>
> >
> >
>
3)(Product[(i^j)*(j^i),{i,1,n},{j,1,n}]/Product[i!*j!,{i,1,n},{j,1,n}])^(1/n)=
> > Product[C(n,i),{i,0,n}]
>
> I confirmed this for n = 5.
> (The products in the LHS grow very rapidly!)
> (For n=5 the Numerator was 2.37*10^65 and the
> Denominator was 2.43*10^45)
>
> >
> > OEIS A001142
> >
> > This identities are known?
> >
> > Sincerely
> >
> > Sebastian Martin Ruiz
>
> Good job on finding those identities, I personally
> do not know if they
> have been found before or not, someone else will
> have to speak to that.
> I just wanted to let you know about the problem
> with the 2nd Identity
> that you found.
>
> -David C.
>

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• Sebastian Martin escribió: Dear colleagues: I have found two identities: 1) Product[k^Floor[n/k],{k,1,n}]=
Message 2 of 4 , Jan 17, 2007
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Sebastian Martin <sebi_sebi@...> escribió:

Dear colleagues:

I have found two identities:

1) Product[k^Floor[n/k],{k,1,n}]=
Product[Floor[n/k]!,{k,1,n}]

2)(Product[(i^j)*(j^i),{i,1,n},{j,1,n}]/Product[i!*j!,{i,1,n},{j,1,n}])^(1/n)=
Product[C(n,i),{i,0,n}]

OEIS A001142

This identities are known?

Sincerely

Sebastian Martin Ruiz

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