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

1,2

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).

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.

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

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

Cf. A176898, A078531.

Sequence in context: A183242 A186019 A186011 * A281835 A299079 A299842

Adjacent sequences: A281730 A281731 A281732 * A281734 A281735 A281736

Keyword

nonn

Author

Felix Fröhlich, Jan 31 2017

Extensions

Extended by Davin Park, Feb 06 2017

Status

approved