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A263547 E.g.f. satisfies: A(x) = exp( x * real( A(x)^I ) ), where I^2 = -1.

%I A263547 #5 Oct 20 2015 22:20:40
%S A263547 1,1,1,-2,-11,36,421,-1896,-35223,201232,5188201,-35856160,
%T A263547 -1188970595,9633456704,391498316301,-3636762088064,-175238714193967,
%U A263547 1835360835895552,102369229796454481,-1193179646751072768,-75645902492063337659,971018266973866894336,68985480327663686993141,-966900537026209266460672
%N A263547 E.g.f. satisfies: A(x) = exp( x * real( A(x)^I ) ), where I^2 = -1.
%F A263547 E.g.f.: A(x) = exp( Series_Reversion( x/cos(x) ) ).
%e A263547 E.g.f.: A(x) = 1 + x + x^2/2! - 2*x^3/3! - 11*x^4/4! + 36*x^5/5! + 421*x^6/6! - 1896*x^7/7! - 35223*x^8/8! + 201232*x^9/9! + 5188201*x^10/10! +...
%e A263547 where
%e A263547 log(A(x)) = x - 3*x^3/3! + 65*x^5/5! - 3787*x^7/7! + 427905*x^9/9! - 79549811*x^11/11! +...+ A036778(n)*x^(2*n-1)/(2*n-1)! +...
%e A263547 which equals Series_Reversion( x/cos(x) ).
%e A263547 Also,
%e A263547 A(x)^I = 1 + I*x - x^2 - 4*I*x^3 + 13*x^4 + 96*I*x^5 - 541*x^6 - 5888*I*x^7/7! + 47545*x^8/8! +...+ A058014(n)*I^n*x^n/n! +...
%e A263547 Further,
%e A263547 Series_Reversion(A(x)-1) = log(1+x)/cos(log(1+x)) = e.g.f. of A009424.
%o A263547 (PARI) {a(n) = my(A=1); for(i=1,n+1, A = exp(x*real(A^I) +x*O(x^n))); n!*polcoeff(A,n)}
%o A263547 for(n=0,30,print1(a(n),", "))
%o A263547 (PARI) {a(n) = my(A=1); for(i=1,n+1, A = exp( serreverse( x/cos(x +x*O(x^n))))); n!*polcoeff(A,n)}
%o A263547 for(n=0,30,print1(a(n),", "))
%Y A263547 Cf. A036778, A058014.
%K A263547 sign
%O A263547 0,4
%A A263547 _Paul D. Hanna_, Oct 20 2015