[seqfan] Re: My tool for exploring sequences

[Thomas Baruchel]

On Mon, 21 Dec 2015, Alexander Povolotsky wrote:
> Sure - I appreciate your method and your efforts!

Hi, after these holydays, I would like to wish you a happy new year to all of you.
I want also to take some time to illustrate how to use my tool.

I spent ~5-10 minutes browsing the OEIS database for choosing a sequence.
I finally took A000166 :

     Subfactorial or rencontres numbers, or derangements: number of
     permutations of n elements with no fixed points.
     (Formerly M1937 N0766)
     1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570,
     176214841, 2290792932, ...

There are some more terms, but I exactly copied the first ones which I used with
my tool; I typed:

   java -jar oeis-deconvolution2-1.0.0-standalone.jar \
   -d ../../oeis-deconvolution/stripped.gz \
   "1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, \
   14684570, 176214841, 2290792932"

Here is the output of my program; I give some comments below:

     OEIS/deconvolution by Th. Baruchel
       initial norm = 5279001364251477795
     Deconvolution with A000166:
       is (1 0 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 1
     Deconvolution with A002741:
       is (1 1 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 2
     Deconvolution with A135799:
       is (1 0 1 -2 4 -8 16 -32 64 -128 256 -512 1024 -2048)
       --> norm is 5592406
     Deconvolution with A173184:
       is (1 -1 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 2
     Deconvolution with A257953:
       is (1 0 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 1
     Deconvolution with A260091:
       is (1 0 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 1
     Deconvolution with A260111:
       is (1 0 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 1
     Deconvolution with A260115:
       is (1 0 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 1
     Deconvolution with A260216:
       is (1 0 0 0 0 0 0 0 0 0 0 0 0 0)
       --> norm is 1

First of all, this is the output with the default settings, and it looks not too
bad at all; with other sequences, it may be useful to change the -t setting.

   a) The sequence is detected in the database: A000166 (first result)
   b) A connected sequence A002741 is detected (but A000166 already tells about
      this connection).
   c) The program also detects something marvelous with A135799, but A000166
      doesn't seem to be aware of that; have a closer look...
      The page for A135799 doesn't tell either about A000166, but A135799 depends
      itself of A134832 which doesn't tell about A000166 either...
      Let's go back to A135799 which tells about A000757 and finally we find
      a relation between A000757 and A000166.
      Maybe something more direct could be found?
      My tools computes as a resulting sequence:
        (1 0 1 -2 4 -8 16 -32 64 -128 256 -512 1024 -2048)
        which is obviously related to generating sequence 1/(2*x+1)
        The exact G.f. actually is: 1+x^2/(2*x+1) in case it would be important.
      Thus, if I am not wrong, any G.f. for A000166 divided by 1+x^2/(2*x+1)
      should be a generating function for A135799. There are many complicated
      G.f. for the first one and almost nothing for the second one, maybe
      something interesting for A135799 would be worth being kept?
   d) A connected sequence A173184 is detected; nothing surprising since A173184
      is defined with A000166; maybe it would be worth adding a link to A173184
      from A000166?
   e) A257953 is a sublist of A000166;
   etc.

I may post about another sequence in some time if I find another example, but
I hope it can help to understand the idea...

Regards,

-- 
Thomas Baruchel