[seqfan] more on double factorials/Penson
Karol PENSON
penson at lptl.jussieu.fr
Tue Oct 20 21:06:35 CEST 2009
Dear Seqfans,
I am trying to clarify a little mess stirred by my remark. From what we
see below the ::::::::: line
A) the only sequence which really deserves the name "double factorial
numbers " is A006882.
In this sequence a line should be added
which links it to the doublefactorial function of Maple, i.e.
A006882(n)=doublefactorial(n). This
line is conspicuously absent from A006882.
B) I propose , to avoid any confusion , not to use the term "double
factorial numbers" in referring to
A001147, to A000165 and to A001818, or rather to use the term
"subsequence of A006882" or
"bisection of A006882", (Neil , could you decide ?) .
Thanks to Richard Mathar and Maximilian Hasler for constructive
observations and remarks ,
best,
Karol A. Penson
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
The image below is a copy from A. P. Prudnikov, Y. Brychkov and
O.I.Marichev,
Integrals and
Series, vol.3, More Special Functions
Moscow,
Fizmatlit, 2003 ( in Russian), p.686
(see also English edition: Gordon and Breach, New York - London , 1990)
One could nevertheless add a comment to clarify that the subsequence (2n-1)!!
is the*list* of the*numbers* called "double factorial numbers",
not the*function* double factorial.
(Traditionally, this name is used only for the (odd) double
fact.numbers (esp. in thermodynamics),
probably because the even version is easily simplified to 2^n (n/2)! )
Maximilian
On Tue, Oct 20, 2009 at 7:41 AM, Richard Mathar
<mathar at strw.leidenuniv.nl> wrote:
>
> kap> Return-Path:<seqfan-bounces at list.seqfan.eu>
> kap> Date: Tue, 20 Oct 2009 12:35:40 +0200
> kap> From: Karol<penson at lptl.jussieu.fr>
> kap> To: Sequence Fanatics Discussion list<seqfan at list.seqfan.eu>,penson at lptl.jussieu.fr
> kap> Subject: [seqfan] definition of double factorial
> kap>
> kap> This is to inform you that the definition of double factorial numbers
> kap> as given in A001147(n) and in related sequences as A001818(n) etc.
> kap> disagrees with Maple's
> kap> function doublefactorial(n). I do not know if this point was raised
> kap> before.
> kap> As this may lead to inconsistencies I intend to submit a comment on it.
> kap>
> kap> Karol A.Penson
>
> There is no disagreement. A001147 and A001818 are related to double factorial
> numbers doublefactorial(2*n-1). The definition says "Double factorial
> numbers: (2n-1)!!" and that this is a bisection of the double factorial
> numbers seems to be so obvious from the text (which says 2n-1, not n) that
> this cannot lead to confusion, I think. Again, A000165 also says
> "Double factorial numbers: (2n)!!", and it seems obvious that this is
> the other bisection. To avoid confusion and not to cause panic, I propose
> not to add a comment on this besides adding the Maple programs like
>
> %p A000165 A000165 := proc(n) doublefactorial(2*n) ; end:
>
>
> %p A001147 A001147 := proc(n) doublefactorial(2*n-1) ; end:
>
> %p A001818 A001818 := proc(n) (A001147(n))2 ; end:
>
> because experience in the past has shown that not all Maple users are
> aware of the existence of this Maple function in the standard library.
>
> Richard Mathar
>
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