central trinomial coefficients

[Bruce Eli Sagan]

David Wilson's observation about the central trinomial coefficients
T_n modulo three was proved in a preprint by Emeric Deutsch and myself

    Congruences for Catalan and Motzkin numbers and related sequences

Stimulated by the discussion on this list, he and I looked at the
general prime p case, conjecturing (probably within minutes of David)
and proving the condition for divisibility in terms of the expansion
of n base p.  The proof easily follows from Lucas' congruence for
binomial coefficients which says that if the digits of n and k base p
are n_0, n_1, ... and k_0, k_1, ... repectively then

    {n choose k} = product_i {n_i choose k_i} (mod p).

Details are in the paper which can be found via the preprints link on
my home page 

	  www.math.msu.edu

See in particular Theorem 4.8, Corollary 4.9, and Conjecture 5.7.

I should also note that we have *not* been able to prove the
conjecture that for n<P, T_n is divisible by p iff T_{p-n-1} is.
Lucas' theorem is no help here since both n and p-n-1 have only one
digit in base p.

Best wishes, Bruce Sagan