[seqfan] A261009

[jean-paul allouche]

Dear Seqfans

As recalled in http://www.integers-ejcnt.org/p43/p43.mail.html
[Integers: Powers in prime bases and a problem on central coefficients]
the following problem, cited in "Concrete Mathematics" by
Graham, Knuth, and Patashnik, is still open: prove that for all n > 256
one has that binom(2n,n) is either divisible by 4 or by 9.

This can be easily reduced to: if c_n is the n-th term of A261009,
then for all k \geq 9, one has
2c_k - c_{k+1} \geq 4.

1. This has been proved up to huge values of k (see ref. above where
     k = 10^{13} is indicated). This is certainly (as any base change 
problem)
     a very difficult question... except if some ingenious trick would 
work:
     any idea?

2. This conjecture does not appear in A261009 on the OEIS.

3. The following appears about this sequence on the OEIS:
     "a261009 = a053735 . a000079  -- Reinhard Zumkeller, Aug 14 2015"
     I guess this should read "a261009 = a053735 composed with a000079"
     as indicated a few lines above:
     "a(n) = A053735(A000079(n)). - Michel Marcus, Aug 14 2015"

Best wishes
jean-paul