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

[Vladimir Shevelev]

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