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

1, 4, 24, 224, 2880, 48064, 989184, 24218624, 687083520, 22151148544, 799546834944, 31934834253824, 1398132497448960, 66573473015578624, 3425078687463112704, 189331392774496845824, 11190654534195295027200, 704262689221037166690304, 47015904809670036594622464, 3318579148264602406039322624

Offset: 1

Author

Paul D. Hanna, Aug 27 2018

Keywords

• nonn

Examples

			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! + ...
such that:
cos(A(x)) + sin(A(x)) = 1/( cos(x) - sin(x) ).
RELATED SERIES.
(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! + ...
(b) If F(F(x)) = A(x), then
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! + ...
where F(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A318006, A318000, A200560.

Programs

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)}
for(n=1,20, print1(a(n),", "))

Formula

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

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

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

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

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

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

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