OFFSET
0,3
COMMENTS
Note that the generating function expands into a power series in x with only real integer coefficients.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(3)*i)^n / (1 + sqrt(3)*i*x*(1+x)^n)^(n+1).
G.f.: Sum_{n>=0} x^n * ((1+x)^n - sqrt(3)*i)^n / (1 - sqrt(3)*i*x*(1+x)^n)^(n+1).
EXAMPLE
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 + ...
Let r = sqrt(3)*i, so that r^2 = -3, then
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 + ...
also,
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 + ...
PROG
(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))}
for(n=0, 30, print1(a(n), ", "))
(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))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 12 2019
STATUS
approved