A052132 Numerators of coefficients in function a(x) such that a(a(a(x))) = sin x.

1, -1, -7, -643, -13583, -29957, -24277937, -6382646731, 2027394133729, 10948179003324221, 177623182156029053, 126604967848904128751, -2640658729595838040517543, -423778395125199663867841

Offset

0,3

References

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

Links

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

Formula

a(n) = numerator(T(2*n-1,1)), T(n,m) = 1/3*((((-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(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-T(m,m)*sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 10 2012

Mathematica

n = 14; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[ CoefficientList[Series[a @ 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 04 2011 *)

Prog

(Maxima)
T(n, m):=if n=m then 1 else 1/3*((((-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(k, m)*sum(T(n, i)*T(i, k), i, k, n), k, m+1, n-1)-T(m, m)*sum(T(n, i)*T(i, m), i, m+1, n-1));
makelist(num(T(2*n-1, 1)), n, 1, 7); /* Vladimir Kruchinin, Mar 10 2012 */

Crossrefs

Cf. A052135. See also A048602, A048603, etc.

Apart from signs, same as A052134?

Sequence in context: A246113 A277839 A109542 * A052134 A101811 A092326

Adjacent sequences: A052129 A052130 A052131 * A052133 A052134 A052135

Keyword

sign,frac,easy,nice

Author

N. J. A. Sloane, Jan 22 2000

Extensions

More terms from R. J. Mathar, coded equivalent to A052136 - R. J. Mathar, Dec 09 2009

Status

approved