cp's OEIS Frontend

A048602 Numerators of coefficients in function a(x) such that a(a(x)) = sin(x).

%I A048602 #61 Mar 08 2025 15:13:26
%S A048602 1,-1,-1,-53,-23,-92713,-742031,594673187,329366540401,
%T A048602 104491760828591,1508486324285153,-582710832978168221,
%U A048602 -1084662989735717135537,-431265609837882130202597,784759327625761394688977441
%N A048602 Numerators of coefficients in function a(x) such that a(a(x)) = sin(x).
%C A048602 A recursion exists for coefficients, but is too complicated to process without a computer algebra system.
%D A048602 W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
%D A048602 W. C. Yang, Composition equations, preprint, 1999
%H A048602 Dmitry Kruchinin and Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation A^{2^n}(x) = F(x)</a>, arXiv:1302.1986 [math.CO], 2013.
%H A048602 W. C. Yang, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00412-4">Derivatives are essentially integer partitions</a>, Discrete Math., 222 (2000), 235-245.
%F A048602 a(n) = numerator(T(n,1)) where T(n,m) = if n=m then 1 else ((((-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(T(n,i)*T(i,m), i=m+1..n-1))/2. - _Vladimir Kruchinin_, Nov 08 2011
%F A048602 a(n) = numerator( A098932(n)/(2^(n-1) * (2*n-1)!) ). - _Andrew Howroyd_, Feb 20 2022
%e A048602 x - x^3/12 - x^5/160 ...
%t A048602 n = 15; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[CoefficientList[Series[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}]; (- CoefficientList[a[x] /. Flatten @ Array[s, n], x] // Numerator // Partition[#, 2] &)[[All, 2]] (* _Jean-François Alcover_, May 05 2011 *)
%o A048602 (Maxima)
%o A048602 T(n,m):= if n=m then 1 else ((((-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(n,i)*T(i,m),i,m+1,n-1))/2; makelist(num(T(n,1)),n,1,10); /* _Vladimir Kruchinin_, Nov 08 2011 */
%o A048602 (PARI) a(n) = { my(ps = sin(x + O(x^(2*n))), q=0); while(ps<>q, q=ps; ps=(sin(serreverse(ps)) + ps)/2); numerator(polcoef(ps, 2*n-1)) } \\ _Gottfried Helms_, Feb 20 2022
%Y A048602 Denominators are A048603.
%Y A048602 Apart from signs, the same sequence as A048606.
%Y A048602 Cf. A098932 (normalized version).
%K A048602 frac,sign,nice
%O A048602 0,4
%A A048602 Winston C. Yang (yang(AT)math.wisc.edu)