A323688 - OEIS

%I #4 Feb 12 2019 22:42:14

%S 1,39,4914,1036674,306362988,116429448762,54085355493876,

%T 29694009536806140,18811303988392369116,13506257834309548172844,

%U 10838311025576842625186286,9612978973113439745046522234,9338259937723855837235460233514,9860260989532664468069402304530610,11244403367868838452318619557009699582,13772731580491669420660957058197302913122,18033025522536379279790808472409496887914884

%N G.f.: Sum_{n>=0} 3^n * ((1+x)^n + i)^n / (4 + 3*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} 3^n * ((1+x)^n + i)^n / (4 + 3*i*(1+x)^n)^(n+1).

%F G.f.: Sum_{n>=0} 3^n * ((1+x)^n - i)^n / (4 - 3*i*(1+x)^n)^(n+1).

%F G.f.: Sum_{n>=0} 3^n * ((1+x)^n + i)^n * (4 - 3*i*(1+x)^n)^(n+1) / (16 + 9*(1+x)^(2*n))^(n+1).

%F G.f.: Sum_{n>=0} 3^n * ((1+x)^n - i)^n * (4 + 3*i*(1+x)^n)^(n+1) / (16 + 9*(1+x)^(2*n))^(n+1).

%e G.f.: A(x) = 1 + 39*x + 4914*x^2 + 1036674*x^3 + 306362988*x^4 + 116429448762*x^5 + 54085355493876*x^6 + 29694009536806140*x^7 + 18811303988392369116*x^8 + ...

%e such that

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

%e also

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

%e RELATED INFINITE SERIES.

%e At x = -1/4, the g.f. as a power series in x diverges, but the related series converges:

%e S = Sum_{n>=0} 3^n * ((3/4)^n + i)^n / (4 + 3*i*(3/4)^n)^(n+1).

%e Equivalently,

%e S = Sum_{n>=0} 12^n * (3^n + 4^n*i)^n / (4^(n+1) + 3^(n+1)*i)^(n+1) ;

%e written explicitly,

%e S = 1/(4+3*i) + 12*(3+4*i)/(4^2+3^2*i)^2 + 12^2*(3^2+4^2*i)^2/(4^3+3^3*i)^3

%e + 12^3*(3^3+4^3*i)^3/(4^4+3^4*i)^4 + 12^4*(3^4+4^4*i)^4/(4^5+3^5*i)^5

%e + 12^5*(3^5+4^5*i)^5/(4^6+3^6*i)^6 + 12^6*(3^6+4^6*i)^6/(4^7+3^7*i)^7 + ...

%e which equals the real number

%e S = 0.33582962080420548450017336765358292462921076160047749762227134638...

%o (PARI) {a(n) = my(A = sum(m=0, n*60 + 600, 3^m*((1+x +x*O(x^n))^m + I)^m/(4 + 3*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*60 + 600, 3^m*((1+x +x*O(x^n))^m - I)^m/(4 - 3*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, A323571.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 12 2019