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

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

Offset: 0

Author

N. J. A. Sloane, Jan 22 2000

Keywords

• nonn
• frac
• easy
• nice

References

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

Crossrefs

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

Programs

Maple

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

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 - 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 *)

Formula

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

Extensions

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