A048605 Numerators of coefficients in function a(x) such that a(a(x)) = arctan(x).

1, -1, 7, -43, 4489, -49897, 20130311, -319053131, 329796121169, -62717244921977, 14635852695795623, -33233512260583073, 149490010959849868177, -3562767949848393597053

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 use without a computer algebra system.

Examples

			x - x^3/6 + x^5 * 7/120 + ...
		

References

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

Links

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

Crossrefs

Cf. A048604, A095885.

Programs

Mathematica

n = 28; a[x_] = Sum[c[k] k! x^k, {k, 1, n, 2}];
sa = Series[a[x], {x, 0, n}];
coes = CoefficientList[ComposeSeries[sa, sa] - Series[ArcTan[x], {x, 0, n}], x] // Rest;
eq = Reduce[((# == 0) & /@ coes)]; Table[c[k] k!, {k, 1, n, 2}] /. First[Solve[eq]] // Numerator
(* Jean-François Alcover, Apr 26 2011 *)

Maxima

T(n, m):=if n=m then 1 else 1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum((2^i*stirling1(i, m)*binomial(n-1, i-1))/i!, i, m, n)-sum(T(n, i)*T(i, m), i, m+1, n-1));
makelist(num(T(2*n-1, 1), n, 1, 5)); /* Vladimir Kruchinin, Mar 12 2012 */

Formula

a(n) = numerator(T(2*n-1,1)), T(n,m)=1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum(i=m..n, (2^i*stirling1(i,m)*binomial(n-1,i-1))/i!)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 12 2012