cp's OEIS Frontend

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)!.

%I A098932 #28 Jan 06 2023 20:45:49
%S A098932 1,-1,-3,-53,-1863,-92713,-3710155,594673187,329366540401,
%T A098932 104491760828591,19610322215706989,-5244397496803513989,
%U A098932 -7592640928150019948759,-2156328049189410651012985,3923796638128806973444887205
%N 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)!.
%C A098932 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.
%H A098932 David Broadhurst, <a href="/A098932/b098932.txt">Table of n, a(n) for n = 1..193</a> (first 100 terms from Paul D. Hanna)
%e A098932 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! + ...
%t A098932 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}]]];
%t A098932 Table[Print[n, " ", a[n]]; a[n], {n, 1, 30}] (* _Jean-François Alcover_, Aug 16 2022, after PARI code *)
%o A098932 (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))}
%o A098932 for(n=1,30,print1(a(n),", "))
%Y A098932 Cf. A095883 (inverse).
%K A098932 frac,sign
%O A098932 1,3
%A A098932 Edward Scheinerman (ers(AT)jhu.edu), Oct 20 2004
%E A098932 More terms from _Paul D. Hanna_, Dec 09 2004
%E A098932 Extended b-file from _David Broadhurst_, Jan 05 2023, submitted by _Stan Wagon_