A095883 Let F(x) be the function such that F(F(x)) = arcsin(x), then F(x) = Sum_{n>=0} a(n)/2^n*x^(2n+1)/(2n+1)!.

1, 1, 13, 501, 38617, 4945385, 944469221, 250727790173, 88106527550129, 39555449833828817, 22093952731139969213, 15041143328788464370373, 12273562321018687866908553, 11833097802606125967312406457

Offset: 0

Author

Paul D. Hanna, Jun 10 2004

Keywords

• changed
• sign

Comments

It appears that, if arcsin(x) is changed to arcsinh(x) in the definition, the sequence obtained is the same except alternating in sign: 1, -1, 13, -501, ... - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Jul 16 2009

a(35) is negative. - Vaclav Kotesovec, Jan 06 2023

Examples

			F(x) = x + (1/2)*x^3/3! + (13/2^2)*x^5/5! + (501/2^3)*x^7/7! + (38617/2^4)*x^9/9! + ...
Special values:
F(x)=Pi/6 at x=F(1/2) = 0.51137532057552418592144885355...
F(x)=Pi/4 at x=F(sqrt(2)/2) = 0.74287348600976...
		

Links

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

Crossrefs

Cf. A095882, A095884, A095885.

Programs

Mathematica

a[n_] := Module[{A, B, F}, F = ArcSin[x] + O[x]^(2n+3); A = F; For[i = 0, i <= n, i++, B[x_] = InverseSeries[A, x] // Normal; A = (A + B[F])/2]; 2^n* (2n+1)!*SeriesCoefficient[A, {x, 0, 2n+1}]];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Aug 16 2022, after PARI code *)

PARI

{a(n)=local(A,B,F);F=asin(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*(2*n+1)!*polcoeff(A,2 *n+1,x)}

Formula

a(n)=T(2*n+1,1)*2^n*(2*n+1)!, T(n,m)=if n=m then 1 else 1/2(Co(n,m)-sum(i=m+1..n-1, T(n,i)*T(i,m))), Co(n,m)=T121408(n,m)=(m!*(sum(k=0..n-m, (-1)^((k)/2)*(sum(i=0..k, (2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!))*binomial((n-2)/2,(n-m-k)/2)))*((-1)^(n-m)+1))/2. - Vladimir Kruchinin, Nov 11 2011