Re: factorial sequence proposal

wugi wrote:
I posted this question to the OEIS
http://www.research.att.com/~njas/sequences/
but didn't get an answer.
I could as well try here, so here my question about a proposed
sequence:

<<
I am impressed by the sheer extension of the OEIS database.

I tried to enter the following sequence but I couldn't pass the
preview. The idea of it is simple enough, but as it is about a
parametered sequence (depending on seed n) it's not feasible to fill
in with particular values.

I therefore try this way: I'd like to learn if I'm onto a new
definition or in a trodden trail!

The basic idea is, implementing factorials of factorials, or "powered"
factorials.
So I came to use the notation !, !!, !!!, .... !^k, although I found
out that !! is being used for less exciting (in my view) definitions.

A sequence needs a starting number, or seed n : hence,
Seq = Seq(n), with terms a(k)=Seq(n,k) as follows:
a(0) = n
a(1) = n!
a(2) = a(1)! = (n!)! == n!!
.....
a(k) = a(k-1)! = n(!!...!!) == n(!^k)

Such sequences, in turn, offer scope for further "supersuperfactorial"
definitions :-o)

I'd appreciate to know if this definition is useful as a new entry, or
indeed exists already somewhere (I myself could not find such
definition in the factorial references I checked).

The idea is not really new. People have worked with iterated factorials before.

For example:

http://ioannis.virtualcomposer2000.com/math/hfseries.html

There is a reference given to me by Dave L. Renfro in AMM Problems (which I
can't find right now) which gives bounds for it. Using your notation (and if
memory serves right):

n^^k < n(!^k) < n^^(k+1),

where ^^ is the tetration operator. Perhaps that's why your sequence didn't make
it to OEIS.

Some of my amateur's math' themes I put on my webpages
http://home.scarlet.be/~pin12499/qbComplex.html
http://home.scarlet.be/~pin12499/hypereal.htm
http://home.scarlet.be/~pin12499/paratwin.htm
I'd appreciate an expert's comments or advice on those as well "!"
Best regards
guido
--
I.N. Galidakis

.

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