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

[Charles Greathouse]

Any thoughts about A164003?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University



On Sat, Jul 31, 2010 at 6:15 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
> 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
>> >> >
>> >> >
>> >>
>> >>
>> >>
>> >> _______________________________________________
>> >>
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>> >>
>> >
>> >  Shevelev Vladimir‎
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>>
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>
>  Shevelev Vladimir‎
>
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