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A323572 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n + i)^n / (A(x) + 1 + i*(1+x)^n)^(n+1), where i^2 = -1.

%I A323572 #3 Feb 10 2019 19:35:20
%S A323572 1,5,45,1142,47253,2664573,187170069,15598588065,1497110942013,
%T A323572 162226788530207,19566798092698042,2598785222401424468,
%U A323572 376850999493886187699,59248452153964672923677,10039900576546291696149404,1824412367286993070795917580,353943959915575446954764374094,73024199735586268826145811783169,15966496692824534985042866376857576,3688160965656359052252569464435170928,897528733209823570848685886402050648933
%N A323572 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n + i)^n / (A(x) + 1 + i*(1+x)^n)^(n+1), where i^2 = -1.
%C A323572 It is remarkable that the generating function results in a power series in x with only real coefficients.
%F A323572 G.f. A(x) satisfies:
%F A323572 (1) 1 = Sum_{n>=0} ((1+x)^n + i)^n / (A(x) + 1 + i*(1+x)^n)^(n+1).
%F A323572 (2) 1 = Sum_{n>=0} ((1+x)^n - i)^n / (A(x) + 1 - i*(1+x)^n)^(n+1).
%F A323572 (3) 1 = Sum_{n>=0} ((1+x)^n + i)^n * (A(x) + 1 - i*(1+x)^n)^(n+1) / ((A(x) + 1)^2 + (1+x)^(2*n))^(n+1).
%F A323572 (4) 1 = Sum_{n>=0} ((1+x)^n - i)^n * (A(x) + 1 + i*(1+x)^n)^(n+1) / ((A(x) + 1)^2 + (1+x)^(2*n))^(n+1).
%e A323572 G.f.: A(x) = 1 + 5*x + 45*x^2 + 1142*x^3 + 47253*x^4 + 2664573*x^5 + 187170069*x^6 + 15598588065*x^7 + 1497110942013*x^8 + 162226788530207*x^9 + ...
%e A323572 such that
%e A323572 1 = 1/(A(x) + 1+i) + ((1+x) + i)/(A(x) + 1 + i*(1+x))^2  +  ((1+x)^2 + i)^2/(A(x) + 1 + i*(1+x)^2)^3  +  ((1+x)^3 + i)^3/(A(x) + 1 + i*(1+x)^3)^4  +  ((1+x)^4 + i)^4/(A(x) + 1 + i*(1+x)^4)^5  +  ((1+x)^5 + i)^5/(A(x) + 1 + i*(1+x)^5)^6 + ...
%e A323572 also,
%e A323572 1 = 1/(A(x) + 1-i) + ((1+x) - i)/(A(x) + 1 - i*(1+x))^2  +  ((1+x)^2 - i)^2/(A(x) + 1 - i*(1+x)^2)^3  +  ((1+x)^3 - i)^3/(A(x) + 1 - i*(1+x)^3)^4  +  ((1+x)^4 - i)^4/(A(x) + 1 - i*(1+x)^4)^5  +  ((1+x)^5 - i)^5/(A(x) + 1 - i*(1+x)^5)^6 + ...
%o A323572 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = round( Vec( sum(m=0, #A*20+300, ((1+x+x*O(x^n))^m + I)^m / (Ser(A) + 1 + I*(1+x+x*O(x^n))^m)^(m+1)*1. ) )[#A]) ); A[n+1]}
%o A323572 for(n=0, 20, print1(a(n), ", "))
%o A323572 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = round( Vec( sum(m=0, #A*20+300, ((1+x+x*O(x^n))^m - I)^m / (Ser(A) + 1 - I*(1+x+x*O(x^n))^m)^(m+1)*1. ) )[#A]) ); A[n+1]}
%o A323572 for(n=0, 20, print1(a(n), ", "))
%Y A323572 Cf. A323313.
%K A323572 nonn
%O A323572 0,2
%A A323572 _Paul D. Hanna_, Feb 10 2019