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A318006 E.g.f. A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) ).

%I A318006 #14 Aug 28 2018 12:45:18
%S A318006 1,2,6,40,360,4592,70896,1279360,26497920,619457792,16166151936,
%T A318006 466022394880,14708199367680,504453778491392,18681868054910976,
%U A318006 742996971891097600,31583887537425776640,1429076863804079931392,68575244394079858262016,3478457209493103235563520,185971933231431479545036800
%N A318006 E.g.f. A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) ).
%H A318006 Paul D. Hanna, <a href="/A318006/b318006.txt">Table of n, a(n) for n = 1..300</a>
%F A318006 E.g.f. A(x) satisfies:
%F A318006 (1) A(-A(-x)) = x.
%F A318006 (2a) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(x) - (-1)^n*A(-x) )^n/n!.
%F A318006 (2b) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(A(x)) + (-1)^n*x )^n/n!.
%F A318006 (3a) 1 = cos( A(x) - A(-x) ) + sin( A(x) + A(-x) ).
%F A318006 (3b) 1 = ( cos(A(x)) + sin(A(x)) ) * ( cos(A(-x)) + sin(A(-x)) ).
%F A318006 (4a) 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x).
%F A318006 (4b) 1 = ( cos(A(A(x))) + sin(A(A(x))) ) * (cos(x) - sin(x)).
%F A318006 (5a) A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2.
%F A318006 (5b) A(A(x)) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2, which is the e.g.f. of A318005.
%F A318006 (6) cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).
%F A318006 a(n) ~ sqrt(3) * 5^(1/4) * 2^(n-2) * n^(n-1) / (exp(n) * (arcsin(2/3))^(n - 1/2)). - _Vaclav Kotesovec_, Aug 28 2018
%e A318006 E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 360*x^5/5! + 4592*x^6/6! + 70896*x^7/7! + 1279360*x^8/8! + 26497920*x^9/9! + 619457792*x^10/10! + ...
%e A318006 such that
%e A318006 cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) )
%e A318006 and
%e A318006 sin(2*A(x)) = 2*sin(2*x)/(2 - sin(2*x)) = 2*sin(A(x))*cos(A(x)).
%e A318006 RELATED SERIES.
%e A318006 (a) A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 48064*x^6/6! + 989184*x^7/7! + 24218624*x^8/8! + ... + A318005(n)*x^n/n! + ...
%e A318006 where A(A(x)) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.
%e A318006 (b) cos(A(x)) + sin(A(x)) = 1/(cos(A(-x)) - sin(A(-x))) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! - 59*x^5/5! - 239*x^6/6! - 421*x^7/7! + 4913*x^8/8! + 108361*x^9/9! + 1000321*x^10/10! + ...
%e A318006 where cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).
%t A318006 nmax = 25; Rest[CoefficientList[Series[ArcSin[4/(2 - Sin[2*x]) - 2]/2, {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Aug 28 2018 *)
%o A318006 (PARI) /* A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2 */
%o A318006 {a(n) = my(A = asin( 2*sin(2*x +x*O(x^n))/(2 - sin(2*x +x*O(x^n))) )/2 ); n!*polcoeff(A,n)}
%o A318006 for(n=1,25, print1(a(n),", "))
%o A318006 (PARI) /* From 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x) */
%o A318006 {a(n) = my(A=[1,1]); for(i=1, n, A = concat(A,0);
%o A318006 A[#A] = -Vec(cos(subst(x*Ser(A),x,x*Ser(A)) + x) + sin(subst(x*Ser(A),x,x*Ser(A)) - x))[#A+1]/2; ); n!*A[n]}
%o A318006 for(n=1, 25, print1(a(n), ", "))
%Y A318006 Cf. A318005, A318001, A200560.
%K A318006 nonn
%O A318006 1,2
%A A318006 _Paul D. Hanna_, Aug 27 2018