A359554 -id:A359554 - cp's OEIS frontend

A003712 Expansion of e.g.f. sin(sin(x)) (odd powers only).

1, -2, 12, -128, 1872, -37600, 990784, -32333824, 1272660224, -59527313920, 3252626013184, -204354574172160, 14594815769038848, -1174376539738169344, 105595092426069327872, -10530693390637550272512

Offset: 0

R. H. Hardin, Simon Plouffe

sign

abs(a(n)) has e.g.f. sinh(sinh(x)) (odd powers only).

abs(a(n)) is the number of partitions of the set {1, 2, ..., 2*n-1} into an odd number of blocks, each containing an odd number of elements. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 23 2004

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 6th line of table.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..100 (first 50 terms from T. D. Noe)

Cf. A359553/A359554.

With[{max = 50}, Take[CoefficientList[Series[Sin[Sin[x]], {x, 0, max}], x] Range[0, max - 1]!, {2, -1, 2}]] (* Vincenzo Librandi, Apr 11 2014 *)
Table[Sum[(-1)^(m + n) (1 + 2k - 2m)^(2n + 1)/(4^k (1 + 2k - m)! m!), {k, 0, n}, {m, 0, k + 1/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)

a(n):=sum((sum((2*i-2*j+1)^(2*n+1)*binomial(2*j-1,i)*(-1)^(n-i-1),i,0,(2*j-1)/2)/(4^(j-1)*(2*j-1)!)),j,1,n+1); /* Vladimir Kruchinin, Jun 09 2011 */

a(n) = Sum_{j=1..n+1} (1/(4^(j-1)*(2*j-1)!)) * Sum_{i=0..(2*j-1)/2} (2*i-2*j+1)^(2*n+1) * binomial(2*j-1,i)*(-1)^(n-i-1). - Vladimir Kruchinin, Jun 09 2011

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

Original entry on oeis.org

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

Offset: 0

Kevin Ryde, Jan 09 2023

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)).

			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 + ...

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.

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

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

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)).

cp's OEIS Frontend

A003712 Expansion of e.g.f. sin(sin(x)) (odd powers only).

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