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%I #22 Sep 12 2015 11:00:19
%S 1,2,12,112,1440,23552,467712,10926592,293544960,8914583552,
%T 301957742592,11285975498752,461367611228160,20477098870833152,
%U 980591931131953152,50393637174029320192,2766350676943951626240
%N E.g.f. satisfies: A(x) = x + sin( A(x) )^2 with A(0)=0.
%C Radius of convergence of A(x) is r = Pi/4 - 1/2, with A(r) = Pi/4.
%H Vaclav Kotesovec, <a href="/A143134/b143134.txt">Table of n, a(n) for n = 1..370</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of implicit functions if Fww = 0</a>, Jan 19 2014
%F E.g.f. satisfies:
%F (1) A(x) = Series_Reversion( x - sin(x)^2 ).
%F (2) A(x) = x + Sum_{n>=1} (-1)^(n-1)*2^(2*n-1) * A(x)^(2*n)/(2*n)!.
%F (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x)^(2*n)/n!.
%F (4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (sin(x)^(2*n)/x)/n! ).
%F E.g.f. derivative: A'(x) = 1/(1 - 2*sqrt(A(x)-x)*sqrt(1+x-A(x))); thus A'(x) = 1/(1 - sin(2*A(x))).
%F Let f(x) = 1/(1-sin(2*x)). Then a(n) = (f(x)*d/dx)^(n-1) f(x) evaluated at x = 0. - Peter Bala, Oct 12 2011
%F a(n) ~ GAMMA(1/3) * 2^(2*n-3/2) * n^(n-5/6) / (3^(1/6) * sqrt(Pi) * exp(n) * (Pi-2)^(n-1/3)). - _Vaclav Kotesovec_, Jan 18 2014
%e A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
%e sin(A(x)) = G(x) is the e.g.f. of A143135:
%e G(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1261*x^5/5! +...
%e G(x)^2 = 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
%e Related expansions:
%e A(x) = x + sin(x)^2 + d/dx sin(x)^4/2! + d^2/dx^2 sin(x)^6/3! + d^3/dx^3 sin(x)^8/4! +...
%e log(A(x)/x) = sin(x)^2/x + d/dx (sin(x)^4/x)/2! + d^2/dx^2 (sin(x)^6/x)/3! + d^3/dx^3 (sin(x)^8/x)/4! +...
%t Rest[CoefficientList[InverseSeries[Series[x - Sin[x]^2,{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 18 2014 *)
%o (PARI) {a(n)=local(A=x);for(i=0,n,A=x + sin(A)^2);n!*polcoeff(A,n)}
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-sin(x+x*O(x^n))^2),n)}
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, sin(x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(A, n)}
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, sin(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A143135, A143136, A214770.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 27 2008
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