A380555 E.g.f. A(x) satisfies A(x) = log( 1 + x * cos(2*A(x)) ).

1, -1, -10, 90, 364, -17760, 85280, 5447120, -116082720, -1709304480, 123520217600, -637137072000, -136024779843200, 3988924415257600, 131963952741504000, -11250603940363008000, 19125068757338752000, 28119635304260378112000, -943657308179458552576000, -59184868918118854443520000

Offset

1,3

Links

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

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) exp(A(x)) = 1 + x * cos(2*A(x)).

(2) A(x) = log( 1 + x * cos(2*A(x)) ).

(3) A( (exp(x) - 1)/cos(2*x) ) = x.

(4) A(x) = Series_Reversion( (exp(x) - 1)/cos(2*x) ).

Example

E.g.f.: A(x) = x - x^2/2! - 10*x^3/3! + 90*x^4/4! + 364*x^5/5! - 17760*x^6/6! + 85280*x^7/7! + 5447120*x^8/8! - 116082720*x^9/9! - 1709304480*x^10/10! + ...
SPECIAL VALUES.
A(t) = Pi/16 at t = 0.2348273750777024091348769029539035346094...
  where t = (exp(Pi/16) - 1) * 2/sqrt(2 + sqrt(2)).
A(t) = Pi/20 at t = 0.1788419348189777972181354090557549056970...
  where t = (exp(Pi/20) - 1) * sqrt(2/5) * sqrt(5 - sqrt(5)).
A(t) = Pi/24 at t = 0.1447869365509419179517924812606040896260...
  where t = (exp(Pi/24) - 1) * sqrt(2) * (sqrt(3) - 1).
A(t) = Pi/32 at t = 0.1051765042663303122710070527373480540972...
  where t = (exp(Pi/32) - 1) * 2/sqrt(2 + sqrt(2 + sqrt(2))).
SPECIFIC VALUES.
A(1/4) = 0.20622896305658178490114810132496023364946226486284...
  where A(1/4) = log( 1 + (1/4)*cos(2*A(1/4)) ).
A(1/5) = 0.17245713659793550611733876887712582250401979882536...
  where A(1/5) = log( 1 + (1/5)*cos(2*A(1/5)) ).
A(1/6) = 0.14792487411803287676006534562611718490228530287793...
  where A(1/6) = log( 1 + (1/6)*cos(2*A(1/6)) ).
A(1/8) = 0.11485990457955157002021678730408576231050832334885...
  where A(1/8) = log( 1 + (1/8)*cos(2*A(1/8)) ).

Prog

(PARI) {a(n) = my(X = x + x*O(x^n)); n!*polcoef( serreverse( (exp(X) - 1)/cos(2*X) ), n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A380055, A380556.

Sequence in context: A024132 A192898 A044261 * A065690 A202919 A202576

Adjacent sequences: A380552 A380553 A380554 * A380556 A380557 A380558

Keyword

sign

Author

Paul D. Hanna, Jan 28 2025

Status

approved