OFFSET
0,4
COMMENTS
A recursion exists for coefficients, but is too complicated to process without a computer algebra system.
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.
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
EXAMPLE
x - x^3/12 - x^5/160 ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
frac,sign,nice
AUTHOR
Winston C. Yang (yang(AT)math.wisc.edu)
STATUS
approved