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%I #25 Sep 02 2023 11:29:35
%S 1,9,135,25515,45927,12629925,4433103675,1396427657625,23739270179625,
%T 21920842083865725,34525326282088516875,8734907549368394769375,
%U 17688187787470999407984375,413903594226821386146834375
%N Denominators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x).
%D W. C. Yang, Composition equations, preprint, 1999.
%F a(x) = Sum_{n>=0} (A052136(n)/A052137(n))*x^(2n+1). - _R. J. Mathar_, Jun 21 2007
%p 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
%t n = 14; m = 2 n - 1 (* m = maximal degree *);
%t 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 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}]);
%t Table[T[2*n - 1, 1] // Denominator, {n, 1, 14}] (* _Jean-François Alcover_, Jul 13 2016, after _Vladimir Kruchinin_ *)
%Y Cf. A052136. See also A048602, A048603, etc.
%K nonn,frac,easy,nice
%O 0,2
%A _N. J. A. Sloane_, Jan 22 2000
%E More terms from _R. J. Mathar_, Jun 21 2007
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