A052136 - OEIS

%I #26 Jul 14 2016 08:35:27

%S 1,-1,4,-263,181,-19417,2650183,-334415182,2505796264,-1075533383968,

%T 644250947168711,-35934792935656882,59703596150692742866,

%U -2784264154855168826899,13245106337447512956269,145404446885533849363819862,-576405412549008975387674250194

%N Numerators 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=0,1,2,3...} A052136(n)/A052137(n)*x^(2n+1). - _R. J. Mathar_, Jun 21 2007

%F 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]

%p 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

%t n = 17; m = 2n - 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] // Numerator // 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] // Numerator, {n, 1, 17}] (* _Jean-François Alcover_, Jul 13 2016, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o 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));

%o makelist(num(T(2*n-1,1)),n,1,7); /* _Vladimir Kruchinin_, Mar 10 2012 */

%Y Cf. A052137. See also A048602, A048603, etc.

%K sign,frac,easy,nice

%O 0,3

%A _N. J. A. Sloane_, Jan 22 2000

%E More terms from _R. J. Mathar_, Jun 21 2007