OFFSET
0,2
COMMENTS
On Apr 19 2011, Zhi-Wei Sun conjectured that a(n) is an integer for every n=0,1,2,.... He proved that a(p-1)=[(p+1)/6] (mod p) for any prime p, and also made the following conjecture:
(i) a(n)^{1/n} tends to 64 as n tends to the infinity.
(ii) For any positive integer n, we have a(n)=0 (mod 8), and a(n)/8 is odd if and only if n is a power of two.
FORMULA
Recursion: (n+2)^2*(3n+2)(3n+4)(3n+5)a(n+2)
=16(2n+1)(2n+3)(3n+2)(18n^2+54n+41)a(n+1) - 9216(n+1)^2(4n^2-1)(3n+5)a(n).
EXAMPLE
For n=1 we have a(1)=(C(6,3)C(3,1)+C(6,3)C(3,1))/C(3,1)=120/3=40.
MATHEMATICA
S[n_]:=Sum[Binomial[6k, 3k]Binomial[3k, k]Binomial[3(n-k), n-k]Binomial[6(n-k), 3(n-k)], {k, 0, n}]/((2n-1)Binomial[3n, n])
Table[S[n], {n, 0, 19}]
CROSSREFS
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Apr 19 2011
STATUS
approved