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 A052132 Numerators of coefficients in function a(x) such that a(a(a(x))) = sin x. 2

%I

%S 1,-1,-7,-643,-13583,-29957,-24277937,-6382646731,2027394133729,

%T 10948179003324221,177623182156029053,126604967848904128751,

%U -2640658729595838040517543,-423778395125199663867841

%N Numerators of coefficients in function a(x) such that a(a(a(x))) = sin x.

%D W. C. Yang, Composition equations, preprint, 1999.

%F a(n)=numerator(T(2*n-1,1)), T(n,m)=1/3*((((-1)^(n-m)+1)*sum(i=0..m/2, (2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i)))/(2^m*n!)-sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-T(m,m)*sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 10 2012

%t n = 14; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[ CoefficientList[Series[a @ a @ a @ x - Sin[x], {x, 0, m}], x] // Rest , 0]; Do[s[k] = Solve[coes[[1]] == 0] // First; coes = coes /. s[k] // Rest, {k, 1, n}]

%t (CoefficientList[a[x] /. Flatten @ Array[s, n], x] // Numerator // Partition[#, 2] &)[[All, 2]]

%t (* _Jean-François Alcover_, May 04 2011 *)

%o (Maxima)

%o T(n,m):=if n=m then 1 else 1/3*((((-1)^(n-m)+1)*sum((2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i),i,0,m/2))/(2^m*n!)-sum(T(k,m)*sum(T(n,i)*T(i,k),i,k,n),k,m+1,n-1)-T(m,m)*sum(T(n,i)*T(i,m),i,m+1,n-1));

%o makelist(num(T(2*n-1,1)),n,1,7); [From _Vladimir Kruchinin_, Mar 10 2012]

%Y Apart from signs, same as A052134?

%K sign,frac,easy,nice

%O 0,3

%A _N. J. A. Sloane_, Jan 22 2000

%E More terms from R. J. Mathar, coded equivalent to A052136 - _R. J. Mathar_, Dec 09 2009

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