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A318005 E.g.f.: A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)).

%I A318005 #9 Aug 27 2018 12:50:40
%S A318005 1,4,24,224,2880,48064,989184,24218624,687083520,22151148544,
%T A318005 799546834944,31934834253824,1398132497448960,66573473015578624,
%U A318005 3425078687463112704,189331392774496845824,11190654534195295027200,704262689221037166690304,47015904809670036594622464,3318579148264602406039322624
%N A318005 E.g.f.: A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)).
%H A318005 Paul D. Hanna, <a href="/A318005/b318005.txt">Table of n, a(n) for n = 1..300</a>
%F A318005 E.g.f. A(x) satisfies:
%F A318005 (1) A(-A(-x)) = x.
%F A318005 (2) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(x) + (-1)^n*x )^n/n!.
%F A318005 (3a) 1 = cos(A(x) + x) + sin(A(x) - x).
%F A318005 (3b) 1 = ( cos(A(x)) + sin(A(x)) ) * ( cos(x) - sin(x) ).
%F A318005 (4) A(x) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2.
%F A318005 a(n) = 2^(n-1) * A200560(n).
%e A318005 E.g.f.: 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! + 687083520*x^9/9! + 22151148544*x^10/10! + ...
%e A318005 such that:
%e A318005 cos(A(x)) + sin(A(x)) = 1/( cos(x) - sin(x) ).
%e A318005 RELATED SERIES.
%e A318005 (a) cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! + ... + A001586(n)*x^n/n! + ...
%e A318005 (b) If F(F(x)) = A(x), then
%e A318005 F(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! + ... + A318006(n)*x^n/n! + ...
%e A318005 where F(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.
%o A318005 (PARI) {a(n) = my(A = asin( sin(2*x +x*O(x^n))/(1 - sin(2*x +x*O(x^n))) )/2 ); n!*polcoeff(A,n)}
%o A318005 for(n=1,20, print1(a(n),", "))
%Y A318005 Cf. A318006, A318000, A200560.
%K A318005 nonn
%O A318005 1,2
%A A318005 _Paul D. Hanna_, Aug 27 2018