A359553 Numerator of the coefficient of x^(2n+1) in the Taylor series expansion of sin(sin(x)).

1, -1, 1, -8, 13, -47, 15481, -15788, 451939, -23252857, 186846623, -831520891, 1108990801, -143356511198507, 920716137922619, -13390469094133441, 929480267163260699, -118186323448146684881, 69875813865886026036091, -155759565768613453511731

Offset: 0

Author

Kevin Ryde, Jan 09 2023

Keywords

• sign
• frac
• easy

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)).

Examples

			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 + ...
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..220
• Christopher Towse, Iteration of Sine and Related Power Series, Mathematics Magazine, volume 87, number 5, December 2014, pages 338-349.

Crossrefs

Cf. A359554 (denominators), A003712 (e.g.f. sin(sin(x))).

Programs

PARI

a_vector(len) = apply(numerator, Vec(substpol(sin(sin(Ser('x,,2*len)))/'x, 'x^2,'x)));

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)).