login
E.g.f.: A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)).
4

%I #9 Aug 27 2018 12:50:40

%S 1,4,24,224,2880,48064,989184,24218624,687083520,22151148544,

%T 799546834944,31934834253824,1398132497448960,66573473015578624,

%U 3425078687463112704,189331392774496845824,11190654534195295027200,704262689221037166690304,47015904809670036594622464,3318579148264602406039322624

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

%H Paul D. Hanna, <a href="/A318005/b318005.txt">Table of n, a(n) for n = 1..300</a>

%F E.g.f. A(x) satisfies:

%F (1) A(-A(-x)) = x.

%F (2) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(x) + (-1)^n*x )^n/n!.

%F (3a) 1 = cos(A(x) + x) + sin(A(x) - x).

%F (3b) 1 = ( cos(A(x)) + sin(A(x)) ) * ( cos(x) - sin(x) ).

%F (4) A(x) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2.

%F a(n) = 2^(n-1) * A200560(n).

%e 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 such that:

%e cos(A(x)) + sin(A(x)) = 1/( cos(x) - sin(x) ).

%e RELATED SERIES.

%e (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 (b) If F(F(x)) = A(x), then

%e 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 where F(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.

%o (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 for(n=1,20, print1(a(n),", "))

%Y Cf. A318006, A318000, A200560.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 27 2018