A323683 - OEIS

%I #10 Feb 12 2019 16:08:12

%S 1,1,2,11,34,147,741,3723,20622,122611,765147,5039624,34856671,

%T 251799740,1895620770,14834210103,120368044792,1010721096231,

%U 8766998042793,78426199305392,722487492991540,6845217410165959,66620876210526469,665308346568565094,6810637828466836635,71400562836982319210,765934679671944152100

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

%C Note that the generating function expands into a power series in x with only real integer coefficients.

%H Paul D. Hanna, <a href="/A323683/b323683.txt">Table of n, a(n) for n = 0..300</a>

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

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

%e G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 34*x^4 + 147*x^5 + 741*x^6 + 3723*x^7 + 20622*x^8 + 122611*x^9 + 765147*x^10 + 5039624*x^11 + 34856671*x^12 + ...

%e Let r = sqrt(3)*i, so that r^2 = -3, then

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

%e also,

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

%o (PARI) {a(n) = my(r = sqrt(3)*I, A = sum(m=0,n+1, x^m*((1+x +x*O(x^n))^m + r)^m/(1 + r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A,n))}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = my(r = sqrt(3)*I, A = sum(m=0,n+1, x^m*((1+x +x*O(x^n))^m - r)^m/(1 - r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A,n))}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A323681, A323682, A323684, A323685.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 12 2019