OFFSET
1,2
COMMENTS
The terms are given on page 3 in Sun (2013).
Conjecture: T_p == -2 (mod p) for any prime p (cf. Sun (2013), Conjecture 4).
LINKS
Davin Park, Table of n, a(n) for n = 1..100
K. H. Pilehrood and T. H. Pilehrood, Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients, The Journal of Integer Sequences, 18 (2015), Article 15.11.7.
Z. W. Sun, Products and sums divisible by central binomial coefficients, The Electronic Journal of Combinatorics, 20(1) (2013), #P9.
FORMULA
a(n) = 16^(n-1) * binomial(3*n-2, 2*n-1)/n. - Sarah Selkirk, Feb 11 2020
From Stefano Spezia, Feb 11 2020: (Start)
O.g.f.: (1/24)*(1 - cos((2/3) * arcsin(6 * sqrt(3*x)))).
E.g.f.: (1/24)*(1 - F([-1/3, 1/3], [1/2, 1], 108*x)), where F is the generalized hypergeometric function. (End)
MATHEMATICA
CoefficientList[Series[(1/24)(1 - Cos[(2/3) ArcSin[6 Sqrt[3x]]]), {x, 0, 20}], x] // Rest (* Davin Park, Feb 06 2017, updated by Jean-François Alcover, Mar 21 2020 *)
CoefficientList[Series[(1-HypergeometricPFQ[{-1/3, 1/3}, {1/2, 1}, 108x])/24, {x, 0, 16}], x]*Table[n!, {n, 0, 16}] (* Stefano Spezia, Mar 21 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Fröhlich, Jan 31 2017
EXTENSIONS
Extended by Davin Park, Feb 06 2017
STATUS
approved