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A048605 Numerators of coefficients in function a(x) such that a(a(x)) = arctan(x). 2
1, -1, 7, -43, 4489, -49897, 20130311, -319053131, 329796121169, -62717244921977, 14635852695795623, -33233512260583073, 149490010959849868177, -3562767949848393597053 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

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

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

LINKS

Table of n, a(n) for n=0..13.

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.

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

EXAMPLE

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

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-Fran├žois Alcover, Apr 26 2011 *)

PROG

(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

CROSSREFS

Cf. A048604, A095885.

Sequence in context: A015463 A177507 A258182 * A165210 A162454 A203210

Adjacent sequences:  A048602 A048603 A048604 * A048606 A048607 A048608

KEYWORD

frac,sign,nice

AUTHOR

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

STATUS

approved

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Last modified May 23 14:53 EDT 2017. Contains 286925 sequences.