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 A052136 Numerators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x). 3
 1, -1, 4, -263, 181, -19417, 2650183, -334415182, 2505796264, -1075533383968, 644250947168711, -35934792935656882, 59703596150692742866, -2784264154855168826899, 13245106337447512956269, 145404446885533849363819862, -576405412549008975387674250194 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES W. C. Yang, Composition equations, preprint, 1999. LINKS FORMULA a(x)=sum_{n=0,1,2,3...} A052136(n)/A052137(n)*x^(2n+1). - R. J. Mathar, Jun 21 2007 a(n)=numerator(T(2*n-1,1)), T(n,m)=1/3*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum(i=m..n, (2^i*stirling1(i,m)*binomial(n-1,i-1))/i!)-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] MAPLE interface(labeling=false) : a := 0 : mPow := 17 : 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, ", numer(alph[2*i+1])) ; ; od: # R. J. Mathar, Jun 21 2007 MATHEMATICA n = 17; m = 2n - 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] // Numerator // 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] // Numerator, {n, 1, 17}] (* Jean-François Alcover, Jul 13 2016, after Vladimir Kruchinin *) PROG (Maxima) T(n, m):=if n=m then 1 else 1/3*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum((2^i*stirling1(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)); makelist(num(T(2*n-1, 1)), n, 1, 7); /* Vladimir Kruchinin, Mar 10 2012 */ CROSSREFS Cf. A052137. See also A048602, A048603, etc. Sequence in context: A112982 A061788 A203839 * A089667 A119008 A108134 Adjacent sequences:  A052133 A052134 A052135 * A052137 A052138 A052139 KEYWORD sign,frac,easy,nice AUTHOR N. J. A. Sloane, Jan 22 2000 EXTENSIONS More terms from R. J. Mathar, Jun 21 2007 STATUS approved

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