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%I #11 Feb 12 2019 22:39:07
%S 1,5,90,2782,120746,6745934,460888652,37224433556,3469644818024,
%T 366563797094652,43286560860984450,5649978863795503006,
%U 807731962396091271434,125520224606849421561430,21066610381145458077037434,3797680233697346232224683014,731834222599438183659030765632,150129114941670825877326504381416,32664492662327616894474980982618994,7513192417898597095586779634504947174,1821542500014252695637241514145496199576
%N G.f.: Sum_{n>=0} ((1+x)^n + i)^n / (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} ((1+x)^n + i)^n / (2 + i*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} ((1+x)^n - i)^n / (2 - i*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} ((1+x)^n + i)^n * (2 - i*(1+x)^n)^(n+1) / (4 + (1+x)^(2*n))^(n+1).
%F G.f.: Sum_{n>=0} ((1+x)^n - i)^n * (2 + i*(1+x)^n)^(n+1) / (4 + (1+x)^(2*n))^(n+1).
%e G.f.: A(x) = 1 + 5*x + 90*x^2 + 2782*x^3 + 120746*x^4 + 6745934*x^5 + 460888652*x^6 + 37224433556*x^7 + 3469644818024*x^8 + 366563797094652*x^9 + ...
%e such that
%e A(x) = 1/(2+i) + ((1+x) + i)/(2 + i*(1+x))^2 + ((1+x)^2 + i)^2/(2 + i*(1+x)^2)^3 + ((1+x)^3 + i)^3/(2 + i*(1+x)^3)^4 + ((1+x)^4 + i)^4/(2 + i*(1+x)^4)^5 + ((1+x)^5 + i)^5/(2 + i*(1+x)^5)^6 + ((1+x)^6 + i)^6/(2 + i*(1+x)^6)^7 + ...
%e also
%e A(x) = 1/(2-i) + ((1+x) - i)/(2 - i*(1+x))^2 + ((1+x)^2 - i)^2/(2 - i*(1+x)^2)^3 + ((1+x)^3 - i)^3/(2 - i*(1+x)^3)^4 + ((1+x)^4 - i)^4/(2 - i*(1+x)^4)^5 + ((1+x)^5 - i)^5/(2 - i*(1+x)^5)^6 + ((1+x)^6 - i)^6/(2 - i*(1+x)^6)^7 + ...
%e RELATED INFINITE SERIES.
%e At x = -1/2, the g.f. as a power series in x diverges, but the related series converges:
%e S = Sum_{n>=0} (1/2^n + i)^n / (2 + i/2^n)^(n+1).
%e Equivalently,
%e S = Sum_{n>=0} 2^n * (1 + 2^n*i)^n / (2^(n+1) + i)^(n+1) ;
%e written explicitly,
%e S = 1/(2+i) + 2*(1+2*i)/(2^2+i)^2 + 2^2*(1+2^2*i)^2/(2^3+i)^3
%e + 2^3*(1+2^3*i)^3/(2^4+i)^4 + 2^4*(1+2^4*i)^4/(2^5+i)^5
%e + 2^5*(1+2^5*i)^5/(2^6+i)^6 + 2^6*(1+2^6*i)^6/(2^7+i)^7 + ...
%e which equals the real number
%e S = 0.51940468012345114278789093541014909064793642824169972277862...
%o (PARI) {a(n) = my(A = sum(m=0, n*10 + 400, ((1+x +x*O(x^n))^m + I)^m/(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*10 + 400, ((1+x +x*O(x^n))^m - I)^m/(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. A323571, A323688.
%Y Cf. A323681.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 10 2019
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