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

%I A380055 #11 Jan 25 2025 04:36:42
%S A380055 1,-1,-1,18,-86,-210,8840,-80080,-266220,19991520,-274725100,
%T A380055 -1006434000,123657316600,-2328145274000,-8148732243600,
%U A380055 1621702497792000,-39454300872662000,-113331522571488000,38748502249144766000,-1172806114215446464000,-2126467491228525900000,1525200888587905488960000
%N A380055 E.g.f. satisfies A(x) = log( 1 + x*cos(A(x)) ).
%H A380055 Paul D. Hanna, <a href="/A380055/b380055.txt">Table of n, a(n) for n = 1..300</a>
%F A380055 E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
%F A380055 (1) A(x) = log( 1 + x*cos(A(x)) ).
%F A380055 (2) exp(A(x)) = 1 + x*cos(A(x)) = the e.g.f. of A380057.
%F A380055 (3) A( (exp(x) - 1)/cos(x) ) = x.
%F A380055 (4) A(x) = Series_Reversion( (exp(x) - 1)/cos(x) ).
%e A380055 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! + ...
%e A380055 where exp(A(x)) = 1 + x*cos(A(x)).
%e A380055 RELATED SERIES.
%e A380055 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! + ...
%e A380055 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! + ...
%e A380055 (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! + ...
%e A380055 where A( (exp(x) - 1)/cos(x) ) = x.
%e A380055 SPECIAL VALUES.
%e A380055 A(t) = Pi/10 at t = 0.388102877484829654642911664361938906648...
%e A380055   where t = (exp(Pi/10) - 1) * sqrt(2/5) * sqrt(5 - sqrt(5)).
%e A380055 A(t) = Pi/12 at t = 0.309822824437268302125213263289623021697...
%e A380055   where t = (exp(Pi/12) - 1) * sqrt(2) * (sqrt(3) - 1).
%e A380055 A(t) = Pi/16 at t = 0.221202550489384066264418133743972067830...
%e A380055   where t = (exp(Pi/16) - 1) * 2/sqrt(2 + sqrt(2 + sqrt(2))).
%e A380055 SPECIFIC VALUES.
%e A380055 A(1/3) = 0.27803485275982692914601139870101331536340692581609...
%e A380055   where A(1/3) = log( 1 + (1/3)*cos(A(1/3)) ).
%e A380055 A(1/4) = 0.21838208334344020266585373465948828480095045415275...
%e A380055   where A(1/4) = log( 1 + (1/4)*cos(A(1/4)) ).
%e A380055 A(1/5) = 0.17963608403094093235073066420053332711004361034830...
%e A380055   where A(1/5) = log( 1 + (1/5)*cos(A(1/5)) ).
%e A380055 A(1/6) = 0.15249154388154448186532142316863060968215944337447...
%e A380055   where A(1/6) = log( 1 + (1/6)*cos(A(1/6)) ).
%o A380055 (PARI) {a(n) = my(X = x + x*O(x^n)); n!*polcoef( serreverse( (exp(X) - 1)/cos(X) ), n)}
%o A380055 for(n=1,25,print1(a(n),", "))
%Y A380055 Cf. A380057, A380053.
%K A380055 sign
%O A380055 1,4
%A A380055 _Paul D. Hanna_, Jan 24 2025