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)!.
1, -1, 7, -129, 4489, -249485, 20130311, -2233371917, 329796121169, -62717244921977, 14635852695795623, -4021254983530551833, 1345410098638648813593, -602107783524378517901957, 304930961594040255078182951, -81321478646591075063015099621
Offset: 0
Keywords
Examples
F(x) = x - 1*x^3/3! + 7*x^5/5! - 129*x^7/7! + 4489*x^9/9! - 249485*x^11/11! +- ... Special values: F(x)=Pi/8 at x=F(sqrt(2)-1) = 0.40303074376779286039477674... F(x)=Pi/6 at x=F(sqrt(3)/3) = 0.54854553000356...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..100
Programs
-
Maxima
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)); makelist((2*n-1)!*T(2*n-1,1),n,1,5); /* Vladimir Kruchinin, Mar 12 2012 */
-
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)}
Formula
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