A095883 -id:A095883 - cp's OEIS frontend

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

Paul D. Hanna, Jun 11 2004

sign

			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...

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

Cf. A095882, A095883, A095884, A048605.

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 */

{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)}

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

A098932 Numerators in the power series of a function f such that f(f(x)) = sin(x) where f(x) = Sum_{n>=1} a(n)/2^(n-1)*x^(2n-1)/(2n-1)!.

Original entry on oeis.org

1, -1, -3, -53, -1863, -92713, -3710155, 594673187, 329366540401, 104491760828591, 19610322215706989, -5244397496803513989, -7592640928150019948759, -2156328049189410651012985, 3923796638128806973444887205

Offset: 1

Edward Scheinerman (ers(AT)jhu.edu), Oct 20 2004

Write f(x) = Sum_{k>=0} b(k)*x^k/k!. Take b(0)=0 and b(1)=1. The remaining b(k) can be found by equating coefficients in f(f(x)) == sin(x). Only the odd terms are nonzero. The sequence given here contains the numerators of the series formed by multiplying (2j+1)!*2^j by the j-th odd term.

			f(x) = x - (1/2)*x^3/3! - (3/2^2)*x^5/5! - (53/2^3)*x^7/7! - (1863/2^4)*x^9/9! + ...

David Broadhurst, Table of n, a(n) for n = 1..193 (first 100 terms from Paul D. Hanna)

Cf. A095883 (inverse).

a[n_] := a[n] = Module[{A, B, F}, F = Sin[x+O[x]^(2n+1)]; A = F; For[i = 0, i <= 2n-1, i++, B[x_] = InverseSeries[A, x] // Normal; A = (A+B[F])/2]; If[n<1, 0, 2^(n-1) (2n-1)! SeriesCoefficient[A, {x, 0, 2n-1}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 30}] (* Jean-François Alcover, Aug 16 2022, after PARI code *)

{a(n)=local(A,B,F);F=sin(x+O(x^(2*n+1)));A=F; for(i=0,2*n-1,B=serreverse(A);A=(A+subst(B,x,F))/2); if(n<1,0,2^(n-1)*(2*n-1)!*polcoeff(A,2*n-1,x))}
for(n=1,30,print1(a(n),", "))

More terms from Paul D. Hanna, Dec 09 2004

Extended b-file from David Broadhurst, Jan 05 2023, submitted by Stan Wagon

cp's OEIS Frontend

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)!.

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