OFFSET
1,1
COMMENTS
The following general conjecture implies that a(n) is always an integer.
Conjecture: Let a and b be nonnegative integers, and let n be a positive integer. If a == b (mod 2), then sum_{k=0}^{n-1}(2k+1)*C(n-1,k)^a*C(-n-1,k)^b == 0 (mod n^2); if a - b is odd, then 2*sum_{k=0}^{n-1}(-1)^k*(2k+1)C(n-1,k)^a*C(-n-1,k)^b == 0 (mod n^2).
An extension of the conjecture was proved in the latest version of arXiv:1408.5381. - Zhi-Wei Sun, Sep 14 2014
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..150
Zhi-Wei Sun, Two new kinds of numbers and related divisibility results, arXiv:1408.5381, 2014.
FORMULA
Recurrence (obtained via the Zeilberger algorithm):
-n*(n-2)*(20*n^2+50*n+29)*a(n) - (220*n^4+550*n^3+369*n^2+59*n-10)*a(n+1)
+ (n+2)^2*(20*n^2+10*n-1)*a(n+2) = 0.
EXAMPLE
a(2) = 5 since 2/2^2*( sum_{k=0,1} (2k+1)*C(1,k)^2*C(2+k,k) ) = 1/2*(1+3*3) = 5.
MATHEMATICA
a[n_] := Sum[(2 k + 1)Binomial[n - 1, k]^2 Binomial[n + k, k], {k, 0, n - 1}] 2/n^2
Table[a[n], {n, 1, 25}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 29 2014
STATUS
approved