[seqfan] Re: Even or odd, true or false?

[M. F. Hasler]

On Tue, Feb 26, 2019 at 9:19 AM Peter Luschny wrote:

>       A(n, k) = Hypergeometric2F1([-k, k + 1], [-k - 1], n).
> Conjecture: A(n, k) is odd if and only if n is even or
> (n is odd and k + 2 = 2^j for some j > 0).
>

You could have helped us with some information which you knew or could have
found out easily.
For example, if you enter at wolframalpha.com :
 Hypergeometric2F1( -k, k + 1; -k - 1; n) with k = 2
 Hypergeometric2F1( -k, k + 1; -k - 1; n) with k = 3
 Hypergeometric2F1( -k, k + 1; -k - 1; n) with k = 4
then you find that these are polynomials in n of degree k whose
coefficients are given by A009766, i.e.,
A(n,k) = sum_{ j=0 .. k } A009766(k,j) n^j
The first column (j=0) of A009766 is always 1, i.e.,
A(n,k) = 1 + n B(n,k)  where B(n,k) is a polynomial in n of degree k-1,
whose coefficients are given in A115126 <http://oeis.org/A115126>.
So it's obvious that A(n,k) is odd if n is even, and
if n is odd, then A(n,k) is odd iff the row sum of A115126 is even.
In A115126 it's written that these row sums are given by A000108(k+1) - 1
So your conjecture boils down to:
A000108(k+1) is odd iff k + 2 = 2^j for some j > 0.
In A000108 is written : Deutsch and Sagan prove the Catalan number C_n is
odd if and only if n = 2^a - 1 for some nonnegative integer a.
QED.

- Maximilian