%I #17 Feb 12 2019 21:46:09

%S 1,1,2,7,22,87,377,1771,9026,49199,284983,1745336,11246563,75956728,

%T 535909242,3938660615,30078439304,238154159543,1951238032473,

%U 16514089454284,144148618179948,1295871420550063,11982543274136961,113830968212019730,1109755421437926323,11092205946446644962,113562177701272805808,1189885690276586123039,12749384941695403919951,139593699183914764551501,1560760177586802637547293

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

%C Note that the generating function expands to a power series in x consisting of only real coefficients.

%H Paul D. Hanna, <a href="/A323681/b323681.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: Sum_{n>=0} x^n*((1+x)^n + i)^n / (1 + i*x*(1+x)^n)^(n+1).

%F G.f.: Sum_{n>=0} x^n*((1+x)^n - i)^n / (1 - i*x*(1+x)^n)^(n+1).

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 22*x^4 + 87*x^5 + 377*x^6 + 1771*x^7 + 9026*x^8 + 49199*x^9 + 284983*x^10 + 1745336*x^11 + 11246563*x^12 + ...

%e such that

%e A(x) = 1/(1 + i*x) + x*((1+x) + i)/(1 + i*x*(1+x))^2 + x^2*((1+x)^2 + i)^2/(1 + i*x*(1+x)^2)^3 + x^3*((1+x)^3 + i)^3/(1 + i*x*(1+x)^3)^4 + x^4*((1+x)^4 + i)^4/(1 + i*x*(1+x)^4)^5 + x^5*((1+x)^5 + i)^5/(1 + i*x*(1+x)^5)^6 + x^6*((1+x)^6 + i)^6/(1 + i*x*(1+x)^6)^7 + x^7*((1+x)^7 + i)^7/(1 + i*x*(1+x)^7)^8 + ...

%e also,

%e A(x) = 1/(1 - i*x) + x*((1+x) - i)/(1 - i*x*(1+x))^2 + x^2*((1+x)^2 - i)^2/(1 - i*x*(1+x)^2)^3 + x^3*((1+x)^3 - i)^3/(1 - i*x*(1+x)^3)^4 + x^4*((1+x)^4 - i)^4/(1 - i*x*(1+x)^4)^5 + x^5*((1+x)^5 - i)^5/(1 - i*x*(1+x)^5)^6 + x^6*((1+x)^6 - i)^6/(1 - i*x*(1+x)^6)^7 + x^7*((1+x)^7 - i)^7/(1 - i*x*(1+x)^7)^8 + ...

%e RELATED INFINITE SERIES.

%e At x = -1/2, the g.f. A(x=-1/2) diverges, but the related series converges:

%e S = Sum_{n>=0} (-1/2)^n * (1/2^n + i)^n / (1 - i/2^(n+1))^(n+1).

%e Equivalently,

%e S = Sum_{n>=0} (-1)^n * 2^(n+1) * (1 + 2^n*i)^n / (2^(n+1) - i)^(n+1) ;

%e written explicitly,

%e S = 2/(2-i) - 2^2*(1+2*i)/(2^2-i)^2 + 2^3*(1+2^2*i)^2/(2^3-i)^3

%e - 2^4*(1+2^3*i)^3/(2^4-i)^4 + 2^5*(1+2^4*i)^4/(2^5-i)^5

%e - 2^6*(1+2^5*i)^5/(2^6-i)^6 + 2^7*(1+2^6*i)^6/(2^7-i)^7 + ...

%e which equals the real number

%e S = 0.61999741931719746274134412657304059740143377356135821449819330...

%o (PARI) {a(n) = my(A = sum(m=0,n+1, x^m*((1+x +x*O(x^n) )^m + I)^m/(1 + I*x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A,n)}

%o for(n=0,35,print1(a(n),", "))

%o (PARI) {a(n) = my(A = sum(m=0,n+1, x^m*((1+x +x*O(x^n) )^m - I)^m/(1 - I*x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A,n)}

%o for(n=0,35,print1(a(n),", "))

%Y Cf. A323680, A323682, A323683, A323684, A323685.

%Y Cf. A323570.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 11 2019