|
|
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).
|
|
2
|
|
|
1, 32, 1792, 122880, 9371648, 763363328, 65028489216, 5722507051008, 516147694796800, 47463855386787840, 4433247375867248640, 419423751734223175680, 40109816011998942461952, 3870915577031009050296320, 376519953782381735485374464, 36874663860751966094632157184
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
a(n) = 16^(n-1) * binomial(3*n-2, 2*n-1)/n. - Sarah Selkirk, Feb 11 2020
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
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|