[seqfan] Re: Number of n-digit numbers the binary expansion of which contains k runs of 1's

[Vladimir Shevelev]

Yes, I am ready to do this instead of these new sequences which are thus should be removed.

Best regards,
Vladimir



----- Original Message -----
From: Charles Greathouse <charles.greathouse at case.edu>
Date: Saturday, July 31, 2010 0:26
Subject: [seqfan] Re: Number of n-digit numbers the binary expansion of which contains k runs of 1's
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> These new sequences are exact duplicates of existing 
> sequences.  If
> someone searches for them, they'll find the existing sequences first
> even if we keep these sequences in.  The usual way to 
> handle this is
> to delete the duplicate and move its description and/or comments into
> the new sequence as comments.
> 
> Along the same lines, I recently found that A164003 is a 
> duplicate of
> A005563 (though I can't show that this happens for all rows of A163280
> -- for example, A028552 and A164004 differ on one point; higher
> sequences will diverge further).  Now that's a trickier 
> case, because
> A164003 is slightly more established.  Should it be changed 
> to a
> ,dead, sequence?  Deleted outright?  Kept with 
> forwarding comments?  I
> tend to favor the first... any opinions?
> 
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
> 
> On Fri, Jul 30, 2010 at 4:26 PM, Vladimir Shevelev 
> <shevelev at bgu.ac.il> wrote:
> > Sorry, but you do not give a proof of your statement (at 
> least, I do not see the clearness of it). On the other hand, 
> using generating functions, one can easily prove that, indeed, 
> the number of n-digit binary numbers  containing k runs of 1's 
> equals to C(n,2*k-1), while the number of n-digit binary numbers 
>  containing k runs of 0's equals to C(n,2*k) (in the latter case 
>  we suppose that (n,k) differs from (1,1)). But it is not a 
> combinatirial proof. But, in my opinion, the  identities
> > Sum{k-1,n-k}C(i,k-1)*C(n-i-1,k-1)=C(n,2*k-1),  Sum{i=k,n-k}C(i-
> 1,k-1)*C(n-i,k)= C(n,2*k)
> > do not give a cause to delete these sequences. Indeed, if 
> somebody is interested to see a behavier of binary numbers with 
> this point of view, then he over first terms easily find these 
> sequences with the corresponding names, if they are in OEIS; 
> further, in coments he will see all formulas. I think that these 
> are sufficiently important characteristics of binary numbers 
> which are not described in OEIS, and  not all, which are 
> interested in binary numbers, are able to obtain such results 
> immediately. Please, try  to find these descriptions without 
> these sequences. You  obtain nothing ( in the best case you can 
> see tables of the  binomial cofficients without the first zeros).
> >  Of course, I do not insist on the publication of them. I 
> think that it is a prerogative of Neil.
> >
> > Regards,
> > Vladimir
> >
> >
> >
> > ----- Original Message -----
> > From: Alois Heinz <heinz at hs-heilbronn.de>
> > Date: Friday, July 30, 2010 19:40
> > Subject: [seqfan] Re: Number of n-digit numbers the binary 
> expansion of which contains k runs of 1's
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >
> >> Both new sequences (A179867, A179868) should be deleted.
> >>
> >> C(n,5) and C(n,6) here gives the number of ways to choose the
> >> positions
> >> of the
> >> leftmost 1 (leftmost 0) of the first, second, third run of 1's
> >> (0's) of
> >> the n digit
> >> binary number.
> >>
> >> Alois
> >>
> >> Charles Greathouse schrieb:
> >> > Sum{i=2,n-3})C(i,2)*C(n-i-1,2) is just binomial(n, 5) =
> >> > n*(n-1)*(n-2)*(n-3)*(n-4)/120.  So assuming the formula
> >> is correct,
> >> > a(12) needs to be corrected and the keyword easy should be added.
> >> >
> >> > Charles Greathouse
> >> > Analyst/Programmer
> >> > Case Western Reserve University
> >> >
> >> >
> >>
> >>
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> >  Shevelev Vladimir‎
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> 
> 
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 Shevelev Vladimir‎