A048606 -id:A048606 - cp's OEIS frontend

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

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

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

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

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

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.

Denominators are A048603.
Apart from signs, the same sequence as A048606.
Cf. A098932 (normalized version).

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 *)

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 */

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

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

A048603 Denominators of coefficients in function a(x) such that a(a(x)) = sin x.

Original entry on oeis.org

1, 12, 160, 40320, 71680, 1277337600, 79705866240, 167382319104000, 91055981592576000, 62282291409321984000, 4024394214140805120000, 5882770031248492462080000, 9076273762497674084352000000

Offset: 0

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

Also denominators of coefficients in function a(x) such that a(a(x)) = sinh x.

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

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

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

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.

Cf. A048602, A048606.

n = 13; 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] // Denominator // Partition[#, 2] &)[[All, 2]]
(* Jean-François Alcover, May 05 2011 *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

cp's OEIS Frontend

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

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