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

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

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

Example

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...

Prog

(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)}
(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 */

Crossrefs

Cf. A095882, A095883, A095884, A048605.

Sequence in context: A002614 A333561 A187237 * A361369 A361367 A332050

Adjacent sequences: A095882 A095883 A095884 * A095886 A095887 A095888

Keyword

sign

Author

Paul D. Hanna, Jun 11 2004

Status

approved