A052136 -id:A052136 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 22 2000

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

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

Apart from signs, same as A052134?

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 *)

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 */

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

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

A052137 Denominators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x).

Original entry on oeis.org

1, 9, 135, 25515, 45927, 12629925, 4433103675, 1396427657625, 23739270179625, 21920842083865725, 34525326282088516875, 8734907549368394769375, 17688187787470999407984375, 413903594226821386146834375

Offset: 0

N. J. A. Sloane, Jan 22 2000

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

Cf. A052136. See also A048602, A048603, etc.

interface(labeling=false) : a := 0 : mPow := 15 : for i from 0 to mPow do a := a+alph[2*i+1]*x^(2*i+1) ; od: a2 := 0 : for i from 0 to mPow do a2 := a2+alph[2*i+1]*a^(2*i+1) ; od: a2 := taylor(a2,x=0,2*mPow+2) : a2 := convert(a2,polynom) : a3 := 0 : for i from 0 to mPow do a3 := a3+alph[2*i+1]*a2^(2*i+1) ; od: for i from 0 to mPow do tanCoef[2*i+1] := coeftayl(arctan(x),x=0,2*i+1) ; od: a3 := taylor(a3,x=0,2*mPow+2) : a3 := convert(a3,polynom) : for i from 0 to mPow do tozer := coeftayl(a3,x=0,2*i+1) : alph[2*i+1] := op(1,[solve(tozer=tanCoef[2*i+1],alph[2*i+1])]) : printf("%d, ",denom(alph[2*i+1])) ; ; od: # R. J. Mathar, Jun 21 2007

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 - ArcTan[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 16 2011 *)
T[n_, n_] = 1; T[n_, m_] := T[n, m] = 1/3*(2^(-m - 1)*m!*((-1)^(n + m) + 1)*(-1)^((3*n + m)/2)*Sum[2^i*StirlingS1[i, m]*Binomial[n - 1, i - 1]/i!, {i, 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}]);
Table[T[2*n - 1, 1] // Denominator, {n, 1, 14}] (* Jean-François Alcover, Jul 13 2016, after Vladimir Kruchinin *)

a(x) = Sum_{n>=0} (A052136(n)/A052137(n))*x^(2n+1). - R. J. Mathar, Jun 21 2007

More terms from R. J. Mathar, Jun 21 2007

cp's OEIS Frontend

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

Views

Author

Keywords

References

Crossrefs

Programs

Mathematica

Maxima

Formula

Extensions