# successive differences of A001147 Double factorial numbers

franktaw at netscape.net franktaw at netscape.net
Fri Jul 11 03:04:36 CEST 2008

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
...

webonfim at bol.com.br said:

(quote)
The number of bracelets can be given by a sum involving the Euler totient<br />function, but
in some cases there are very simple expressions that count those<br />objects.<C2><A0><br /><br
/>For example, I found in njas's sequence A001399 that the number of bracelets<br />with n+3
<C2><A0>similar expressions for the case of 4 "reds", or "5 reds"?<br /><br
/>There is<C2><A0>a method to find simple expressions?</p>
(end)

%D A001399 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3\
}.

Neil

HERE'S MY RESPONSE TO the following message:
WITH SOME WORDS IN CAPS TO MAKE the comments visible
NEIL
> Subject: Re: Partitions invariant under a permutation
> Date: Thu, 10 Jul 2008 20:56:11 -0400
> From: franktaw at netscape.net
>
> Motzkin was in fact looking at the same problem area as I am -- so I guess I
> will undertake updating the "Sorting Numbers" sequences after all.
>
> A mistake was made in putting these sequences into the OEIS: the e.g.f.
>
> e^{e^x-1+(e^{px}-1)/p}
>
> given in (6') on page 171 in fact applies only when p is prime.

OF COURSE I REALIZED THIS WHEN I added those sequences!
What I did was to to take Motzkin's g.f.
e^{e^x-1+(e^{px}-1)/p}
which he used when p was a prime, and used the same formula with p a nonprime.
This was not a mistake.  It says very clearly in A036078 that the e.g.f. is
E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=8.
And I called them "sorting numbers", why not?

ON THE OTHER HAND, I agree that your formula,
> e^{Sum_{d|l} (e^{d*x}-1)/d}.
A036074, A036076, A036078, A036079 A036080, and A036082
as new sequences - will you please send them in in the usual way?

> This
> is explicit in equation (6) on page 170, which (6') refers to (it is the
> "corresponding recurrence").  So sequences A036074, A036076, A036078, A036079,
> A036080, and A036082 are in the OEIS under "false pretenses".
>
> As noted in my previous post, the correct general form for this e.g.f.  is
>
> e^{Sum_{d|l} (e^{d*x}-1)/d}.
>
> Motzkin does not give this general form.  I'm wondering if this result is
> published anywhere.  I don't have any way to do this kind of literature
> search at this time.  (I would think that searching references to Motzkin's
> article would cover it.)

ONE WAY TO DO SUCH A SEARCH IS to work out the first few terms
and look up the sequence in the OEIS!

> If it is not published, I think it would be
> worth a note somewhere -- perhaps in the Journal of Integer Sequences.
>
> -----
> My plan for the "Sorting Numbers" sequences is as follows, in outline:
>
> Add the array (by anti-diagonals) T(n,k); the number of partitions of n*k
> objects invariant under a permutation with n k-cycles; which Motzkin writes
> as !^{<u>cy</u>k\dot n} -- pardon the bastardized notation.  This will
> have references to the new (see below) and existing column sequences.
>
GOOD IDEA!

> Change the sequences for prime columns of this array to have a more
> descriptive name, and to reference the array above.  For this and others,
> I would add a comment that this used to be known simply as "Sorting
> Numbers".
YES!
>
> Add new sequences for the composite columns, showing the e.g.f. as
> above.  These would have links to corresponding sequences listed above.
YES!
>
> Change the sequences listed above to have the given e.g.f. as the name;
YES, DONE! NEW VERSION of OEIS later this morning.
> include links to the new column sequences.
YES, will do later
>
> Examine the other sequences titled "Sorting Numbers", finding more
> descriptive names as described by Motzkin.
>
> Modify A084423 (which I recently extended) to reference the new array
> sequence; the U(k,j) in the formula and PARI program is in fact this
> array.
>
> (Neil, I'd especially like your comments on this program -- I want to submit
> the changes as you want them, so that neither of us wastes our time and effort.)
HERE THEY ARE!
>
> Also, add two new "tabf" sequences with the number of partitions
> invariant under a permutation with the index partition as its cycle structure
> (one in Abramowitz and Stegun order and one in Mathematica order) (as I
> described in my previous post).
YES!
>
>
BEST REGARDS

NEIL

Dear Seqfans,  can someone figure out what the definition means?
The link is broken, and A N W Hone has not responded to
my asking for a new URL for the article.
Neil

%I A122046
%S A122046 0,0,1,3,6,10,16,24,34,46,61,79,100,124,152,184,220,260,305,355,410
%N A122046 a(n)=(x^(n - 1)a(n - 1)a(n - 4) + a(n - 2)*a(n - 3))/a(n - 5).
%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.
%H A122046 A. N. W. Hone, <a href="http://www.kent.ac.uk/ims/publications/documents/paper_607.pdf">Algebraic curves, integer sequences and a discrete Painleve transcendent</a>, Proceedings of SIDE 6, Helsinki, Finland, Jun 19 2004. [Set a(n)=d(n+3) on p. 8]
%t A122046 p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1;p[ -4] = 1;p[ -3] = 1;p[ -2] = 1;p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
%Y A122046 Cf. A014125.
%Y A122046 Sequence in context: A121776 A088637 A066377 this_sequence A078663 A025222 A011902
%Y A122046 Adjacent sequences: A122043 A122044 A122045 this_sequence A122047 A122048 A122049
%K A122046 nonn,uned,obsc
%O A122046 0,4
%A A122046 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 13 2006
%E A122046 Partially edited by njas, Sep 17 2006, Jul 11 2008