cp's OEIS Frontend

A143135 E.g.f. satisfies: A(x) = sin(x + A(x)^2) with A(0)=0.

%I A143135 #9 Sep 12 2015 11:00:20
%S A143135 1,2,11,100,1261,20342,399671,9256840,246907321,7452534122,
%T A143135 251099460611,9341422237420,380293239870181,16815919738248542,
%U A143135 802553031266952431,41117164304824602640,2250747364089063475441
%N A143135 E.g.f. satisfies: A(x) = sin(x + A(x)^2) with A(0)=0.
%C A143135 Radius of convergence of A(x) is r = Pi/4 - 1/2, with A(r) = sqrt(2)/2.
%H A143135 Vaclav Kotesovec, <a href="/A143135/b143135.txt">Table of n, a(n) for n = 1..370</a>
%H A143135 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of implicit functions if Fww = 0</a>, Jan 19 2014
%F A143135 E.g.f.: A(x) = sin(G(x)) where G(x) = x + A(x)^2 is the e.g.f. of A143134.
%F A143135 E.g.f. derivative: A'(x) = sqrt(1 - A(x)^2)/(1 - 2*A(x)*sqrt(1 - A(x)^2)).
%F A143135 a(n) ~ GAMMA(1/3) * 4^(n-1) * n^(n-5/6) / (3^(1/6) * sqrt(Pi) * exp(n) * (Pi-2)^(n-1/3)). - _Vaclav Kotesovec_, Jan 19 2014
%e A143135 A(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1261*x^5/5! +...
%e A143135 A(x) = sin(G(x)) where G(x) = x + A(x)^2 is the e.g.f. of A143134:
%e A143135 G(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
%t A143135 Rest[CoefficientList[InverseSeries[Series[-x^2 + ArcSin[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 19 2014 *)
%o A143135 (PARI) {a(n)=local(A=x);for(i=0,n,A=x + sin(A)^2);n!*polcoeff(sin(A),n)}
%o A143135 (PARI) {a(n)=n!*polcoeff(sin(serreverse(x-sin(x+x*O(x^n))^2)),n)}
%Y A143135 Cf. A143134, A143137.
%K A143135 nonn
%O A143135 1,2
%A A143135 _Paul D. Hanna_, Jul 27 2008