cp's OEIS Frontend

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

%I A052132 #24 Jul 04 2018 08:56:18
%S A052132 1,-1,-7,-643,-13583,-29957,-24277937,-6382646731,2027394133729,
%T A052132 10948179003324221,177623182156029053,126604967848904128751,
%U A052132 -2640658729595838040517543,-423778395125199663867841
%N A052132 Numerators of coefficients in function a(x) such that a(a(a(x))) = sin x.
%D A052132 W. C. Yang, Composition equations, preprint, 1999.
%F A052132 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 A052132 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 A052132 (CoefficientList[a[x] /. Flatten @ Array[s, n], x] // Numerator // Partition[#, 2] &)[[All, 2]]
%t A052132 (* _Jean-François Alcover_, May 04 2011 *)
%o A052132 (Maxima)
%o A052132 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 A052132 makelist(num(T(2*n-1,1)),n,1,7); /* _Vladimir Kruchinin_, Mar 10 2012 */
%Y A052132 Cf. A052135. See also A048602, A048603, etc.
%Y A052132 Apart from signs, same as A052134?
%K A052132 sign,frac,easy,nice
%O A052132 0,3
%A A052132 _N. J. A. Sloane_, Jan 22 2000
%E A052132 More terms from _R. J. Mathar_, coded equivalent to A052136 - _R. J. Mathar_, Dec 09 2009