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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)!. 3

%I

%S 1,-1,-3,-53,-1863,-92713,-3710155,594673187,329366540401,

%T 104491760828591,19610322215706989,-5244397496803513989,

%U -7592640928150019948759,-2156328049189410651012985,3923796638128806973444887205

%N 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 Write f[x]=Sum[b[k]x^k/k!,{k,0,Infinity}]. 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 Paul D. Hanna, <a href="/A098932/b098932.txt">Table of n, a(n) for n = 1..100</a>

%e 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! +...

%o (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 for(n=1,30,print1(a(n),", "))

%Y Cf. A095883 (inverse).

%K frac,sign

%O 1,3

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

%E More terms from _Paul D. Hanna_, Dec 09 2004

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Last modified January 16 13:35 EST 2022. Contains 350376 sequences. (Running on oeis4.)