[seqfan] Re: parity of binomials and some carryless products A175669 - A175672

[Antti Karttunen]

On Wed, Aug 11, 2010 at 9:54 PM,  <seqfan-request at list.seqfan.eu> wrote:
> Message: 14
> Date: Wed, 11 Aug 2010 20:25:16 +0200
> From: Richard Mathar <mathar at strw.leidenuniv.nl>
> Subject: [seqfan]  parity of binomials and some carryless products,
>        A175669 - A175672
> To: seqfan at seqfan.eu
> Message-ID: <201008111825.o7BIPGvG019644 at krypton.strw.leidenuniv.nl>
> Content-Type: text/plain; charset=us-ascii
>
>
> These 8 sequences appear highly correlated in pairs:
>
> http://oeis.org/classic/?q=id:A175672|id:A115845
> 0,1,2,3,4,5,6,7,8,10,12,14,16,17,20,21,24,28,32,33,34,35,40,42,48,49,56,64,65,66,67,68,69,70,71,80,81,84,85,96,97,98,99,112,113,128,129,130,131,132,133,134,135,136,138,140,142,160,161,162,163,168,170,192,193,
>
> 0,1,2,3,4,5,6,7,8,10,12,14,16,17,20,21,24,28,32,33,34,35,40,42,48,49,56,64,65,66,67,68,69,70,71,80,81,84,85,96,97,98,99,112,113,128,129,130,131,132,133,134,135,136,138,140,142,160,161,162,163,168,170,192,
>
> http://oeis.org/classic/?q=id:A175671|id:A048715
> 0,1,2,4,8,9,16,17,18,32,33,34,36,64,65,66,68,72,73,128,129,130,132,136,137,144,145,146,256,257,258,260,264,265,272,273,274,288,289,290,292,512,513,514,516,520,521,528,529,530,544,545,546,548,576,577,578,580,
>
> 0,1,2,4,8,9,16,17,18,32,33,34,36,64,65,66,68,72,73,128,129,130,132,136,137,144,145,146,256,257,258,260,264,265,272,273,274,288,289,290,292,512,513,514,516,520,521,528,
>
>
> http://oeis.org/classic/?q=id:A175670|id:A048716
> 0,1,2,3,4,6,8,9,12,16,17,18,19,24,25,32,33,34,35,36,38,48,49,50,51,64,65,66,67,68,70,72,73,76,96,97,98,99,100,102,128,129,130,131,132,134,136,137,140,144,145,146,147,152,153,192,193,194,195,196,198,200,201,
>
> 0,1,2,3,4,6,8,9,12,16,17,18,19,24,25,32,33,34,35,36,38,48,49,50,51,64,65,66,67,68,70,72,73,76,96,97,98,99,100,102,128,129,130,131,132,134,136,137,140,144,145,146,147,
>
> http://oeis.org/classic/?q=id:A175669|id:A003714
> 0,1,2,4,5,8,9,10,16,17,18,20,21,32,33,34,36,37,40,41,42,64,65,66,68,69,72,73,74,80,81,82,84,85,128,129,130,132,133,136,137,138,144,145,146,148,149,160,161,162,164,165,168,169,170,256,257,258,260,261,264,265,
>
> 0,1,2,4,5,8,9,10,16,17,18,20,21,32,33,34,36,37,40,41,42,64,65,66,68,69,72,73,74,80,81,82,84,85,128,129,130,132,133,136,137,138,144,145,146,148,149,160,161,162,164,165,168,169,170,256,257,258,260,
>
> They all deal with parities of binomials C(k*n,n) on one hand
> and the Finnish way of multiplication in GF(2)[x] on the other.

The starving bears ate our carry-bits, so we have to do without...

> It would
> be nice to have some explanations on the mutual correlations
> and --from the purely administrative point of view-- comments inserted
> when they start to differ.

Also, regarding the recent speculation about running all new
submissions through Superseeker:
it would be nice even if it did start with just one transformation:
the identity,
to detect the obvious duplicate-candidates!

BTW, I check every sequence I submit with the python script available from here:
http://oeis.org/w/images/a/a0/User_files-Antti_Karttunen-oeischek_py.txt
it creates a similarly named local HTML-file, where the %S-lines
(after two initial terms)
are search-links to OEIS. (Actually the subsequence to be searched
from OEIS should
be even shorter.)

My hunch is that it shouldn't be too hard to prove that the sequences
mentioned above
are identical. If one considers the parity of binomial coefficient of C(kn,n),
it's the same thing as a bit n on row kn in
http://oeis.org/classic/table?a=47999&fmt=312
(and each row is obtained from the previous row with a simple shifted
xor, see http://oeis.org/classic/A001317 )

But sorry, too tired to do it by myself now.

Cheers,

Antti

>
> Richard Mathar
>