successive differences of A001147 Double factorial numbers

Mitch Harris maharri at gmail.com
Fri Jul 11 16:59:36 CEST 2008

```I personally find much more interest in this suggestion than in the
scads of base and prime sequences. But reflecting on that makes me
think it might be better to be more permissive.

- -everything- has a combinatorial interpretation, just maybe not a
simple one, or one that is already published.

- The first differences of n!! are 2n*n!! and for second differences
the function is  2(2n^2 + 2n + 1)*n!!. Going further, I'm sure there
are interesting patterns in those coefficients (as there are for the
kth-differences for n!).

- not every sequence has to be 'nice' (though I suppose the fewer
there are of 'less' the better)

- my desideratum for 'sequence worth submitting' is if can you have at
least one line in the OEIS for the sequence that is not just
identifying. e.g. a formula, a mathematical relation with another
sequence, a comment, a reference. It'd be preferred for a mathematical
relation to be 'forward' rather than derivative (since the derivative
sequences should be caught be superseeker...ostensibly there's no end
to creating derivative sequences by transforming the results of
previous transforms). What I mean by 'forward' is that from your
sequence apply a transformation, and find that it is equal to one
already known by OEIS, derivative, means applying a transform to one
in the OEIS to get you new sequence. Yes, I realize this is arbitrary
given that transforms have inverses, but motivation does make it more
interesting.

- but to make a triangle worthwhile, I think you'd really have to do
the leg work of computing and submitting all the rows/columns/and-or
antidiagnoals, checking for cross references, and relations to
previous OEIS entries.

Mitch

On Thu, Jul 10, 2008 at 9:04 PM,  <franktaw at netscape.net> wrote:
> My opinion is, unless you have some other reason why this is of interest,
> no.
>
> The double factorials are not of sufficient interest that a simple transform
> of
> them is worth including just on its own -- unlike the factorials, which are
> this
> interesting.  (Besides the fact that A047920 has a combinatorial
> interpretation, which is what (IMO) moves it up from merely "worth
> including"
> to "nice".)  If you were to find a similar combinatorial interpretation of
> your
> sequence, it would certainly be worth including.
>
>
> -----Original Message-----
> From: Jonathan Post <jvospost3 at gmail.com>
>
> Is the following worthy of OEIS submission?  (Or the not shown
> Triangular array formed from successive differences of A006882 Double
> factorials n!!: a(n)=n*a(n-2))?
>
> Analogue of A047920 Triangular array formed from successive
> differences of factorial numbers.
>
> Triangular array formed from successive differences of A001147 Double
> factorial numbers: (2n-1)!! = 1.3.5....(2n-1)..
>
> 0: 1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425
> 1: 0, 2, 12, 90, 840, 9450, 124740, 1891890, 32432400
> ...
>

--
Mitch Harris

njas> From seqfan-owner at ext.jussieu.fr  Fri Jul 11 14:48:15 2008
njas> Date: Fri, 11 Jul 2008 08:46:52 -0400
njas> From: "N. J. A. Sloane" <njas at research.att.com>
njas> To: seqfan at ext.jussieu.fr
njas> Subject: request for help with obscrure sequence
njas> Cc: jerry_metzger at und.nodak.edu, njas at research.att.com
njas>
njas> Dear Seqfans,  can someone figure out what the definition means?
njas> The link is broken, and A N W Hone has not responded to
njas> my asking for a new URL for the article.
njas> Neil
njas> ...
njas> %S A122046 0,0,1,3,6,10,16,24,34,46,61,79,100,124,152,184,220,260,305,355,410
njas> %N A122046 a(n)=(x^(n - 1)a(n - 1)a(n - 4) + a(n - 2)*a(n - 3))/a(n - 5).
njas> %F A122046 Comment from Jerry Metzger (jerry_metzger(AT)und.nodak.edu), Jul 09 2008: It appears that every 4th term is given by Sum_{m=1...n-1} (floor{m(n-2)/n})^2 and the intermediate terms by adding an easy "adjustment" term to that sum.
njas> ...

The definition is
Degree of the polynomial P(n,x)= [x^(n-1)*P(n-1,x)*P(n-4,x)+P(n-2,x)*P(n-3,x)]/P(n-5,x) with P(1,x)=P(0,x)=P(-1,x)=P(-2,x)=P(-3,x)=1.

The sequence continues
0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 61, 79, 100, 124, 152, 184, 220, 260, 305,
355, 410, 470, 536, 608, 686, 770, 861, 959, 1064, 1176, 1296, 1424, 1560, 1704,
1857, 2019, 2190, 2370, 2560

found with a Maple program

P := proc(n) option remember ;
end:
for n from 0 to 80 do
od:

It does not satisfy a linear recurrence with 6 or less coefficients (as
one may have hoped spying at A014125).

Jerry's formula sum_{m=1..n-1} floor[m(n-2)/2]^2 produces the sequence

0, 0
1, 0
2, 0
3, 1
4, 14
5, 62
6, 220
7, 547
8, 1260
9, 2444

generated with

for n from 0 to 15 do
od:

which are hardly related to A122046, because only "1" and "220"
match terms in the sequence (?).

Richard Mathar http://www.strw.leidenuniv.nl/~mathar

```