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%I #9 Feb 12 2019 22:38:38
%S 1,18,1080,109024,15425932,2807373808,624567442464,164229954020384,
%T 49830967023579400,17136446536264628328,6586556828478666923776,
%U 2798143974987211154322560,1301956008571438833028276672,658475124477897905982708920064,359674624553169582199492194921216,211016939776698518257518156591766656,132339913526099601722463496476503304848,88352431866089178236286985348733511803104,62560734523453615823978316851314371901576832
%N G.f.: Sum_{n>=0} 2^n * ((1+x)^n + i)^n / (3 + 2*i*(1+x)^n)^(n+1), where i^2 = -1.
%C It is remarkable that the generating function results in a power series in x with only real coefficients.
%F G.f.: Sum_{n>=0} 2^n * ((1+x)^n + i)^n / (3 + 2*i*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} 2^n * ((1+x)^n - i)^n / (3 - 2*i*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} 2^n * ((1+x)^n + i)^n * (3 - 2*i*(1+x)^n)^(n+1) / (9 + 4*(1+x)^(2*n))^(n+1).
%F G.f.: Sum_{n>=0} 2^n * ((1+x)^n - i)^n * (3 + 2*i*(1+x)^n)^(n+1) / (9 + 4*(1+x)^(2*n))^(n+1).
%e G.f.: A(x) = 1 + 18*x + 1080*x^2 + 109024*x^3 + 15425932*x^4 + 2807373808*x^5 + 624567442464*x^6 + 164229954020384*x^7 + 49830967023579400*x^8 + ...
%e such that
%e A(x) = 1/(3+2*i) + 2*((1+x) + i)/(3 + 2*i*(1+x))^2 + 2^2*((1+x)^2 + i)^2/(3 + 2*i*(1+x)^2)^3 + 2^3*((1+x)^3 + i)^3/(3 + 2*i*(1+x)^3)^4 + 2^4*((1+x)^4 + i)^4/(3 + 2*i*(1+x)^4)^5 + 2^5*((1+x)^5 + i)^5/(3 + 2*i*(1+x)^5)^6 + 2^6*((1+x)^6 + i)^6/(3 + 2*i*(1+x)^6)^7 + ...
%e also
%e A(x) = 1/(3-2*i) + 2*((1+x) - i)/(3 - 2*i*(1+x))^2 + 2^2*((1+x)^2 - i)^2/(3 - 2*i*(1+x)^2)^3 + 2^3*((1+x)^3 - i)^3/(3 - 2*i*(1+x)^3)^4 + 2^4*((1+x)^4 - i)^4/(3 - 2*i*(1+x)^4)^5 + 2^5*((1+x)^5 - i)^5/(3 - 2*i*(1+x)^5)^6 + 2^6*((1+x)^6 - i)^6/(3 - 2*i*(1+x)^6)^7 + ...
%e RELATED INFINITE SERIES.
%e At x = -1/3, the g.f. as a power series in x diverges, but the related series converges:
%e S = Sum_{n>=0} 2^n * ((2/3)^n + i)^n / (3 + 2*i*(2/3)^n)^(n+1).
%e Equivalently,
%e S = Sum_{n>=0} 6^n * (2^n + 3^n*i)^n / (3^(n+1) + 2^(n+1)*i)^(n+1) ;
%e written explicitly,
%e S = 1/(3+2*i) + 6*(2+3*i)/(3^2+2^2*i)^2 + 6^2*(2^2+3^2*i)^2/(3^3+2^3*i)^3
%e + 6^3*(2^3+3^3*i)^3/(3^4+2^4*i)^4 + 6^4*(2^4+3^4*i)^4/(3^5+2^5*i)^5
%e + 6^5*(2^5+3^5*i)^5/(3^6+2^6*i)^6 + 6^6*(2^6+3^6*i)^6/(3^7+2^7*i)^7 + ...
%e which equals the real number
%e S = 0.398373937215203455798934836530015104419943236574659953950499...
%o (PARI) {a(n) = my(A = sum(m=0, n*40 + 400, 2^m*((1+x +x*O(x^n))^m + I)^m/(3 + 2*I*(1+x +x*O(x^n))^m)^(m+1)*1. )); round(polcoeff(A, n))}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = my(A = sum(m=0, n*40 + 400, 2^m*((1+x +x*O(x^n))^m - I)^m/(3 - 2*I*(1+x +x*O(x^n))^m)^(m+1)*1. )); round(polcoeff(A, n))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A323570, A323688.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 10 2019
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