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

1, -1, -1, -53, -23, -92713, -742031, 594673187, 329366540401, 104491760828591, 1508486324285153, -582710832978168221, -1084662989735717135537, -431265609837882130202597, 784759327625761394688977441

Offset: 0

Author

Winston C. Yang (yang(AT)math.wisc.edu)

Keywords

• frac
• sign
• nice

Comments

A recursion exists for coefficients, but is too complicated to process without a computer algebra system.

Examples

			x - x^3/12 - x^5/160 ...
		

References

• W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
• W. C. Yang, Composition equations, preprint, 1999

Links

• Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x) = F(x), arXiv:1302.1986 [math.CO], 2013.
• W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.

Crossrefs

Denominators are A048603.

Apart from signs, the same sequence as A048606.

Cf. A098932 (normalized version).

Programs

Mathematica

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 *)

Maxima

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 */

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

Formula

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

a(n) = numerator( A098932(n)/(2^(n-1) * (2*n-1)!) ). - Andrew Howroyd, Feb 20 2022