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A095885 Let F(x) be the function such that F(F(x)) = arctan(x), then F(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.

%I A095885 #15 Jan 06 2023 10:31:53
%S A095885 1,-1,7,-129,4489,-249485,20130311,-2233371917,329796121169,
%T A095885 -62717244921977,14635852695795623,-4021254983530551833,
%U A095885 1345410098638648813593,-602107783524378517901957,304930961594040255078182951,-81321478646591075063015099621
%N A095885 Let F(x) be the function such that F(F(x)) = arctan(x), then F(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
%H A095885 Vaclav Kotesovec, <a href="/A095885/b095885.txt">Table of n, a(n) for n = 0..100</a>
%F A095885 a(n) = (2*n-1)!*T(2*n-1,1), T(n,m)=1/2*(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(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 12 2012
%e A095885 F(x) = x - 1*x^3/3! + 7*x^5/5! - 129*x^7/7! + 4489*x^9/9! - 249485*x^11/11! +- ...
%e A095885 Special values:
%e A095885 F(x)=Pi/8 at x=F(sqrt(2)-1) = 0.40303074376779286039477674...
%e A095885 F(x)=Pi/6 at x=F(sqrt(3)/3) = 0.54854553000356...
%o A095885 (PARI) {a(n)=local(A,B,F);F=atan(x+x*O(x^(2*n+1)));A=F; for(i=0,n,B=serreverse(A);A=(A+subst(B,x,F))/2);(2*n+1)!*polcoeff(A,2*n+1,x)}
%o A095885 (Maxima)
%o A095885 T(n,m):=if n=m then 1 else 1/2*(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(n,i)*T(i,m),i,m+1,n-1));
%o A095885 makelist((2*n-1)!*T(2*n-1,1),n,1,5); /* _Vladimir Kruchinin_, Mar 12 2012 */
%Y A095885 Cf. A095882, A095883, A095884, A048605.
%K A095885 sign
%O A095885 0,3
%A A095885 _Paul D. Hanna_, Jun 11 2004