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A380556 E.g.f. A(x) satisfies A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.

%I A380556 #10 Jan 29 2025 12:46:29
%S A380556 1,1,0,-12,48,820,-14160,-69160,5900160,-44796960,-3089865600,
%T A380556 88646729600,1412786918400,-135956951062400,1023512450688000,
%U A380556 203887248898944000,-7307555382586368000,-252816835499795840000,26110132266648748032000,-95216226972043640320000,-80962066973581160140800000
%N A380556 E.g.f. A(x) satisfies A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.
%F A380556 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A380556 (1) A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.
%F A380556 (2) A(x) = 1 + x*cos( log( A(x)^2 ) ).
%F A380556 (3) A(x) = exp( L(x) ), where L(x) = Series_Reversion( (exp(x) - 1)/cos(2*x) ) is the e.g.f. of A380555.
%F A380556 (4) A( (exp(x) - 1)/cos(2*x) ) = exp(x).
%e A380556 E.g.f.: A(x) = 1 + x - 12*x^3/3! + 48*x^4/4! + 820*x^5/5! - 14160*x^6/6! - 69160*x^7/7! + 5900160*x^8/8! - 44796960*x^9/9! - 3089865600*x^10/10! + ...
%e A380556  where A(x) = 1 + x*cos( log( A(x)^2 ) ).
%e A380556 RELATED SERIES.
%e A380556 A(x)^(2*i) = 1 + 2*i*x + (-4 - 2*i)*x^2/2! + (12 - 28*i)*x^3/3! + (164 + 228*i)*x^4/4! + (-2360 + 1440*i)*x^5/5! + (-9880 - 51480*i)*x^6/6! + (737520 + 129440*i)*x^7/7! + (-4977440 + 17821440*i)*x^8/8! + (-308986560 - 306791360*i)*x^9/9! + ...
%e A380556  where A(x)^(2*i) = cos( log(A(x)^2) ) + i*sin( log(A(x)^2) ).
%e A380556 L(x) = log(A(x)) = x - x^2 - 10*x^3 + 90*x^4 + 364*x^5 - 17760*x^6 + 85280*x^7 + 5447120*x^8 + ... + A380555(n)*x^n/n! + ...
%e A380556  where L( (exp(x) - 1)/cos(2*x) ) = x.
%e A380556 cos(2*L(x)) = 1 - 4*x^2/2! + 12*x^3/3! + 164*x^4/4! - 2360*x^5/5! - 9880*x^6/6! + 737520*x^7/7! - 4977440*x^8/8! + ... + (a(n+1)/(n+1))*x^n/n! + ...
%e A380556  where A(x) = 1 + x*cos(2*L(x)).
%t A380556 CoefficientList[Exp[InverseSeries[Series[ (Exp[x] - 1)/Cos[2*x] ,{x,0,20}],x]],x]Range[0,20]! (* _Stefano Spezia_, Jan 29 2025 *)
%o A380556 (PARI) {a(n) = my(A = 1+x+x*O(x^n)); for(i=1,n, A = real( 1 + x*A^(2*I) +x*O(x^n) )); n!*polcoef(A,n)}
%o A380556 for(n=0,30, print1(a(n),", "))
%Y A380556 Cf. A380057, A380555.
%K A380556 sign
%O A380556 0,4
%A A380556 _Paul D. Hanna_, Jan 28 2025