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A359553
Numerator of the coefficient of x^(2n+1) in the Taylor series expansion of sin(sin(x)).
3
1, -1, 1, -8, 13, -47, 15481, -15788, 451939, -23252857, 186846623, -831520891, 1108990801, -143356511198507, 920716137922619, -13390469094133441, 929480267163260699, -118186323448146684881, 69875813865886026036091, -155759565768613453511731
OFFSET
0,4
COMMENTS
Denominators are A359554.
Sine is an odd function so the Taylor series has 0 coefficients at even terms x^(2n).
A003712(n) is the numerator for use with denominator (2n+1)! so that here a(n)/A359554(n) = A003712(n)/(2n+1)! reduced to least terms.
abs(a(n)) is the corresponding numerator in the expansion of sinh(sinh(x)).
LINKS
Christopher Towse, Iteration of Sine and Related Power Series, Mathematics Magazine, volume 87, number 5, December 2014, pages 338-349.
FORMULA
a(n) = numerator of A003712(n)/(2n+1)!.
Sum_{n>=0} a(n)/A359554(n) * x^(2*n+1). = sin(sin(x)).
Sum_{n>=0} abs(a(n))/A359554(n) * x^(2*n+1). = sinh(sinh(x)).
EXAMPLE
Fractions begin: 1, -1/3, 1/10, -8/315, 13/2520, -47/49896, ...
Series begins: sin(sin(x)) = x - (1/3)*x^3 + (1/10)*x^5 - (8/315)*x^7 + ...
PROG
(PARI) a_vector(len) = apply(numerator, Vec(substpol(sin(sin(Ser('x, , 2*len)))/'x, 'x^2, 'x)));
CROSSREFS
Cf. A359554 (denominators), A003712 (e.g.f. sin(sin(x))).
Sequence in context: A279711 A043121 A043901 * A081966 A316527 A304546
KEYWORD
sign,frac,easy
AUTHOR
Kevin Ryde, Jan 09 2023
STATUS
approved