A098932 Numerators in the power series of a function f such that f(f(x)) = sin(x) where f(x) = Sum_{n>=1} a(n)/2^(n-1)*x^(2n-1)/(2n-1)!.

1, -1, -3, -53, -1863, -92713, -3710155, 594673187, 329366540401, 104491760828591, 19610322215706989, -5244397496803513989, -7592640928150019948759, -2156328049189410651012985, 3923796638128806973444887205

Offset

1,3

Comments

Write f(x) = Sum_{k>=0} b(k)*x^k/k!. Take b(0)=0 and b(1)=1. The remaining b(k) can be found by equating coefficients in f(f(x)) == sin(x). Only the odd terms are nonzero. The sequence given here contains the numerators of the series formed by multiplying (2j+1)!*2^j by the j-th odd term.

Links

David Broadhurst, Table of n, a(n) for n = 1..193 (first 100 terms from Paul D. Hanna)

Example

f(x) = x - (1/2)*x^3/3! - (3/2^2)*x^5/5! - (53/2^3)*x^7/7! - (1863/2^4)*x^9/9! + ...

Mathematica

a[n_] := a[n] = Module[{A, B, F}, F = Sin[x+O[x]^(2n+1)]; A = F; For[i = 0, i <= 2n-1, i++, B[x_] = InverseSeries[A, x] // Normal; A = (A+B[F])/2]; If[n<1, 0, 2^(n-1) (2n-1)! SeriesCoefficient[A, {x, 0, 2n-1}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 30}] (* Jean-François Alcover, Aug 16 2022, after PARI code *)

Prog

(PARI) {a(n)=local(A, B, F); F=sin(x+O(x^(2*n+1))); A=F; for(i=0, 2*n-1, B=serreverse(A); A=(A+subst(B, x, F))/2); if(n<1, 0, 2^(n-1)*(2*n-1)!*polcoeff(A, 2*n-1, x))}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A095883 (inverse).

Sequence in context: A352992 A012742 A012823 * A100444 A300420 A300683

Adjacent sequences: A098929 A098930 A098931 * A098933 A098934 A098935

Keyword

frac,sign

Author

Edward Scheinerman (ers(AT)jhu.edu), Oct 20 2004

Extensions

More terms from Paul D. Hanna, Dec 09 2004

Extended b-file from David Broadhurst, Jan 05 2023, submitted by Stan Wagon

Status

approved