A380055 E.g.f. satisfies A(x) = log( 1 + x*cos(A(x)) ).

1, -1, -1, 18, -86, -210, 8840, -80080, -266220, 19991520, -274725100, -1006434000, 123657316600, -2328145274000, -8148732243600, 1621702497792000, -39454300872662000, -113331522571488000, 38748502249144766000, -1172806114215446464000, -2126467491228525900000, 1525200888587905488960000

Offset: 1

Author

Paul D. Hanna, Jan 24 2025

Keywords

• sign

Examples

			E.g.f.: A(x) = x - x^2/2! - x^3/3! + 18*x^4/4! - 86*x^5/5! - 210*x^6/6! + 8840*x^7/7! - 80080*x^8/8! - 266220*x^9/9! + 19991520*x^10/10! - 274725100*x^11/11! + ...
where exp(A(x)) = 1 + x*cos(A(x)).
RELATED SERIES.
exp(A(x)) = 1 + x - 3*x^3/3! + 12*x^4/4! + 10*x^5/5! - 660*x^6/6! + 5600*x^7/7! + 8400*x^8/8! - 951660*x^9/9! + 12715200*x^10/10! + ... + A380057(n)*x^n/n! + ...
cos(A(x)) = 1 - x^2/2! + 3*x^3/3! + 2*x^4/4! - 110*x^5/5! + 800*x^6/6! + 1050*x^7/7! - 105740*x^8/8! + 1271520*x^9/9! + ...
(exp(x) - 1)/cos(x) = x + x^2/2! + 4*x^3/3! + 7*x^4/4! + 36*x^5/5! + 91*x^6/6! + 624*x^7/7! + 2087*x^8/8! + ... + A380053(n)*x^n/n! + ...
where A( (exp(x) - 1)/cos(x) ) = x.
SPECIAL VALUES.
A(t) = Pi/10 at t = 0.388102877484829654642911664361938906648...
  where t = (exp(Pi/10) - 1) * sqrt(2/5) * sqrt(5 - sqrt(5)).
A(t) = Pi/12 at t = 0.309822824437268302125213263289623021697...
  where t = (exp(Pi/12) - 1) * sqrt(2) * (sqrt(3) - 1).
A(t) = Pi/16 at t = 0.221202550489384066264418133743972067830...
  where t = (exp(Pi/16) - 1) * 2/sqrt(2 + sqrt(2 + sqrt(2))).
SPECIFIC VALUES.
A(1/3) = 0.27803485275982692914601139870101331536340692581609...
  where A(1/3) = log( 1 + (1/3)*cos(A(1/3)) ).
A(1/4) = 0.21838208334344020266585373465948828480095045415275...
  where A(1/4) = log( 1 + (1/4)*cos(A(1/4)) ).
A(1/5) = 0.17963608403094093235073066420053332711004361034830...
  where A(1/5) = log( 1 + (1/5)*cos(A(1/5)) ).
A(1/6) = 0.15249154388154448186532142316863060968215944337447...
  where A(1/6) = log( 1 + (1/6)*cos(A(1/6)) ).
		

Links

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

Crossrefs

Cf. A380057, A380053.

Programs

PARI

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

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.

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

(2) exp(A(x)) = 1 + x*cos(A(x)) = the e.g.f. of A380057.

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

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