A323571 G.f.: Sum_{n>=0} 2^n * ((1+x)^n + i)^n / (3 + 2*i*(1+x)^n)^(n+1), where i^2 = -1.

1, 18, 1080, 109024, 15425932, 2807373808, 624567442464, 164229954020384, 49830967023579400, 17136446536264628328, 6586556828478666923776, 2798143974987211154322560, 1301956008571438833028276672, 658475124477897905982708920064, 359674624553169582199492194921216, 211016939776698518257518156591766656, 132339913526099601722463496476503304848, 88352431866089178236286985348733511803104, 62560734523453615823978316851314371901576832

Offset

0,2

Comments

It is remarkable that the generating function results in a power series in x with only real coefficients.

Links

Table of n, a(n) for n=0..18.

Formula

G.f.: Sum_{n>=0} 2^n * ((1+x)^n + i)^n / (3 + 2*i*(1+x)^n)^(n+1).

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

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).

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).

Example

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 + ...
such that
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 + ...
also
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 + ...
RELATED INFINITE SERIES.
At x = -1/3, the g.f. as a power series in x diverges, but the related series converges:
S = Sum_{n>=0} 2^n * ((2/3)^n + i)^n / (3 + 2*i*(2/3)^n)^(n+1).
Equivalently,
S = Sum_{n>=0} 6^n * (2^n + 3^n*i)^n / (3^(n+1) + 2^(n+1)*i)^(n+1) ;
written explicitly,
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
+ 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
+ 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 + ...
which equals the real number
S = 0.398373937215203455798934836530015104419943236574659953950499...

Prog

(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))}
for(n=0, 20, print1(a(n), ", "))
(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))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A323570, A323688.

Sequence in context: A146197 A301651 A159379 * A052135 A033518 A333006

Adjacent sequences: A323568 A323569 A323570 * A323572 A323573 A323574

Keyword

nonn

Author

Paul D. Hanna, Feb 10 2019

Status

approved