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%I #9 Jan 12 2023 06:24:19
%S 1,-1,1,-8,13,-47,15481,-15788,451939,-23252857,186846623,-831520891,
%T 1108990801,-143356511198507,920716137922619,-13390469094133441,
%U 929480267163260699,-118186323448146684881,69875813865886026036091,-155759565768613453511731
%N Numerator of the coefficient of x^(2n+1) in the Taylor series expansion of sin(sin(x)).
%C Denominators are A359554.
%C Sine is an odd function so the Taylor series has 0 coefficients at even terms x^(2n).
%C 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.
%C abs(a(n)) is the corresponding numerator in the expansion of sinh(sinh(x)).
%H Kevin Ryde, <a href="/A359553/b359553.txt">Table of n, a(n) for n = 0..220</a>
%H Christopher Towse, <a href="https://www.jstor.org/stable/10.4169/math.mag.87.5.338">Iteration of Sine and Related Power Series</a>, Mathematics Magazine, volume 87, number 5, December 2014, pages 338-349.
%F a(n) = numerator of A003712(n)/(2n+1)!.
%F Sum_{n>=0} a(n)/A359554(n) * x^(2*n+1). = sin(sin(x)).
%F Sum_{n>=0} abs(a(n))/A359554(n) * x^(2*n+1). = sinh(sinh(x)).
%e Fractions begin: 1, -1/3, 1/10, -8/315, 13/2520, -47/49896, ...
%e Series begins: sin(sin(x)) = x - (1/3)*x^3 + (1/10)*x^5 - (8/315)*x^7 + ...
%o (PARI) a_vector(len) = apply(numerator, Vec(substpol(sin(sin(Ser('x,,2*len)))/'x, 'x^2,'x)));
%Y Cf. A359554 (denominators), A003712 (e.g.f. sin(sin(x))).
%K sign,frac,easy
%O 0,4
%A _Kevin Ryde_, Jan 09 2023
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