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

1, 2, 6, 40, 360, 4592, 70896, 1279360, 26497920, 619457792, 16166151936, 466022394880, 14708199367680, 504453778491392, 18681868054910976, 742996971891097600, 31583887537425776640, 1429076863804079931392, 68575244394079858262016, 3478457209493103235563520, 185971933231431479545036800

Offset

1,2

Links

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

Formula

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

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

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

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

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

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

(4a) 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x).

(4b) 1 = ( cos(A(A(x))) + sin(A(A(x))) ) * (cos(x) - sin(x)).

(5a) A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2.

(5b) A(A(x)) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2, which is the e.g.f. of A318005.

(6) cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).

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

Example

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! + ...
such that
cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) )
and
sin(2*A(x)) = 2*sin(2*x)/(2 - sin(2*x)) = 2*sin(A(x))*cos(A(x)).
RELATED SERIES.
(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! + ...
where A(A(x)) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.
(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! + ...
where cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).

Mathematica

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 *)

Prog

(PARI) /* A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2 */
{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)}
for(n=1, 25, print1(a(n), ", "))
(PARI) /* From 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x) */
{a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0);
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]}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A318005, A318001, A200560.

Sequence in context: A277483 A267981 A343846 * A356513 A292407 A274275

Adjacent sequences: A318003 A318004 A318005 * A318007 A318008 A318009

Keyword

nonn

Author

Paul D. Hanna, Aug 27 2018

Status

approved