A095883 - OEIS

%I #24 Jan 06 2023 10:30:23

%S 1,1,13,501,38617,4945385,944469221,250727790173,88106527550129,

%T 39555449833828817,22093952731139969213,15041143328788464370373,

%U 12273562321018687866908553,11833097802606125967312406457

%N 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)!.

%C 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

%C a(35) is negative. - _Vaclav Kotesovec_, Jan 06 2023

%H Vaclav Kotesovec, <a href="/A095883/b095883.txt">Table of n, a(n) for n = 0..100</a>

%F 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

%e 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! + ...

%e Special values:

%e F(x)=Pi/6 at x=F(1/2) = 0.51137532057552418592144885355...

%e F(x)=Pi/4 at x=F(sqrt(2)/2) = 0.74287348600976...

%t 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}]];

%t Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Aug 16 2022, after PARI code *)

%o (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)}

%Y Cf. A095882, A095884, A095885.

%K changed,sign

%O 0,3

%A _Paul D. Hanna_, Jun 10 2004