A281733 Positive integers T_i such that Sum_{k >= 0} (S_k * x^(2*k+1)) + 1/24 - Sum_{k >= 1} (T_k * x^(2*k)) = (cos((2/3) * arccos(6 * sqrt(3) * x)))/12 for all real x with |x| <= 1/(6 * sqrt(3)), where S_k = A176898(k).

1, 32, 1792, 122880, 9371648, 763363328, 65028489216, 5722507051008, 516147694796800, 47463855386787840, 4433247375867248640, 419423751734223175680, 40109816011998942461952, 3870915577031009050296320, 376519953782381735485374464, 36874663860751966094632157184

Offset: 1

Author

Felix Fröhlich, Jan 31 2017

Keywords

• nonn

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).

This is the odd bisection of A078531 divided by 2. The even bisection divided by 2 is A176898. - Akiva Weinberger, Dec 09 2024

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.

Crossrefs

Cf. A176898, A078531.

Programs

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 *)

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)

a(n) = binomial(6n-3, 3n-3/2)*binomial(3n-3/2, n-1/2)/(4*n*binomial(2*n-1, n-1/2)). - Akiva Weinberger, Dec 09 2024

a(n) = A078531(2*n-1)/2. - Akiva Weinberger, Dec 09 2024

Extensions

Extended by Davin Park, Feb 06 2017