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

%I A380555 #11 Jan 29 2025 12:46:16
%S A380555 1,-1,-10,90,364,-17760,85280,5447120,-116082720,-1709304480,
%T A380555 123520217600,-637137072000,-136024779843200,3988924415257600,
%U A380555 131963952741504000,-11250603940363008000,19125068757338752000,28119635304260378112000,-943657308179458552576000,-59184868918118854443520000
%N A380555 E.g.f. A(x) satisfies A(x) = log( 1 + x * cos(2*A(x)) ).
%H A380555 Paul D. Hanna, <a href="/A380555/b380555.txt">Table of n, a(n) for n = 1..300</a>
%F A380555 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380555 (1) exp(A(x)) = 1 + x * cos(2*A(x)).
%F A380555 (2) A(x) = log( 1 + x * cos(2*A(x)) ).
%F A380555 (3) A( (exp(x) - 1)/cos(2*x) ) = x.
%F A380555 (4) A(x) = Series_Reversion( (exp(x) - 1)/cos(2*x) ).
%e A380555 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! + ...
%e A380555 SPECIAL VALUES.
%e A380555 A(t) = Pi/16 at t = 0.2348273750777024091348769029539035346094...
%e A380555   where t = (exp(Pi/16) - 1) * 2/sqrt(2 + sqrt(2)).
%e A380555 A(t) = Pi/20 at t = 0.1788419348189777972181354090557549056970...
%e A380555   where t = (exp(Pi/20) - 1) * sqrt(2/5) * sqrt(5 - sqrt(5)).
%e A380555 A(t) = Pi/24 at t = 0.1447869365509419179517924812606040896260...
%e A380555   where t = (exp(Pi/24) - 1) * sqrt(2) * (sqrt(3) - 1).
%e A380555 A(t) = Pi/32 at t = 0.1051765042663303122710070527373480540972...
%e A380555   where t = (exp(Pi/32) - 1) * 2/sqrt(2 + sqrt(2 + sqrt(2))).
%e A380555 SPECIFIC VALUES.
%e A380555 A(1/4) = 0.20622896305658178490114810132496023364946226486284...
%e A380555   where A(1/4) = log( 1 + (1/4)*cos(2*A(1/4)) ).
%e A380555 A(1/5) = 0.17245713659793550611733876887712582250401979882536...
%e A380555   where A(1/5) = log( 1 + (1/5)*cos(2*A(1/5)) ).
%e A380555 A(1/6) = 0.14792487411803287676006534562611718490228530287793...
%e A380555   where A(1/6) = log( 1 + (1/6)*cos(2*A(1/6)) ).
%e A380555 A(1/8) = 0.11485990457955157002021678730408576231050832334885...
%e A380555   where A(1/8) = log( 1 + (1/8)*cos(2*A(1/8)) ).
%o A380555 (PARI) {a(n) = my(X = x + x*O(x^n)); n!*polcoef( serreverse( (exp(X) - 1)/cos(2*X) ), n)}
%o A380555 for(n=1,25,print1(a(n),", "))
%Y A380555 Cf. A380055, A380556.
%K A380555 sign
%O A380555 1,3
%A A380555 _Paul D. Hanna_, Jan 28 2025