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%I #14 Nov 09 2021 19:53:05
%S 1,1,2,9,43,238,1436,9230,62347,438397,3188687,23886061,183706628,
%T 1447476799,11666912970,96101351315,808489267438,6944890495515,
%U 60909005722973,545480514290103,4989651763986766,46633629693716933,445474644772366480,4351015655544312048,43463615209598548901,444132103382081004621,4642827326458824657758,49649335888812297089157,543030381664721843558408,6072677103962177819391932
%N G.f. A(x) = Sum_{n>=0} x^n*(A(x)^n + i)^n/(1 + i*x*A(x)^n)^(n+1), where i^2 = -1.
%C Notice that the imaginary components in the generating function vanish when expanded as a power series in x.
%H Paul D. Hanna, <a href="/A324619/b324619.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} x^n*(A(x)^n + i)^n / (1 + i*x*A(x)^n)^(n+1).
%F (2) A(x) = Sum_{n>=0} x^n*(A(x)^n - i)^n / (1 - i*x*A(x)^n)^(n+1).
%F From _Paul D. Hanna_, Nov 05 2021: (Start)
%F (3) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} i^k * binomial(n,k) * (A(x)^n - i*A(x)^k)^(n-k).
%F (4) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} (-i)^k * binomial(n,k) * (A(x)^n + i*A(x)^k)^(n-k). (End)
%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 43*x^4 + 238*x^5 + 1436*x^6 + 9230*x^7 + 62347*x^8 + 438397*x^9 + 3188687*x^10 + 23886061*x^11 + 183706628*x^12 + ...
%e such that
%e A(x) = 1/(1 - i*x) + x*(A(x) - i)/(1 - i*x*A(x))^2 + x^2*(A(x)^2 - i)^2/(1 - i*x*A(x)^2)^3 + x^3*(A(x)^3 - i)^3/(1 - i*x*A(x)^3)^4 + x^4*(A(x)^4 - i)^4/(1 - i*x*A(x)^4)^5 + x^5*(A(x)^5 - i)^5/(1 - i*x*A(x)^5)^6 + ...
%e also
%e A(x) = 1/(1 + i*x) + x*(A(x) + i)/(1 + i*x*A(x))^2 + x^2*(A(x)^2 + i)^2/(1 + i*x*A(x)^2)^3 + x^3*(A(x)^3 + i)^3/(1 + i*x*A(x)^3)^4 + x^4*(A(x)^4 + i)^4/(1 + i*x*A(x)^4)^5 + x^5*(A(x)^5 + i)^5/(1 + i*x*A(x)^5)^6 + ...
%o (PARI) {a(n) = my(A=[1,1]); for(i=0,n, A = concat(A,0);
%o A[#A] = polcoeff( sum(n=0,#A, x^n*(Ser(A)^n + I)^n/(1 + I*x*Ser(A)^n)^(n+1) ),#A-1)); polcoeff(Ser(A),n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A324618.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 12 2019
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