OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. satisfies: (1 + x^3*A(x)^3) / (1 + x*A(x)) = (1-x)*(1+x^3).
G.f.: (1 - sqrt(4*(1-x)*(1+x^3) - 3))/(2*x).
a(n) ~ sqrt(1-3*r+r^3) / (2*sqrt(Pi)*n^(3/2)*r^(n+1)), where r = (1-2*sqrt(1/4 + 3^(1/3)/2^(2/3)) + sqrt(2-2*6^(1/3) + 7/sqrt(1/4 + 3^(1/3)/2^(2/3))))/4 = 0.2634705866719... is the root of the equation r^4-r^3+r=1/4. - Vaclav Kotesovec, Dec 08 2013
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 10*x^4 + 29*x^5 + 86*x^6 + 266*x^7 +...
such that, by definition,
1-x = 1 - x*A(x) + x^2*A(x)^2 - x^3 + x^4*A(x) - x^5*A(x)^2 + x^6 - x^7*A(x) + x^8*A(x)^2 - x^9 + x^10*A(x) - x^11*A(x)^2 + ....
MATHEMATICA
CoefficientList[Series[(1 - Sqrt[4*(1-x)*(1+x^3) - 3])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 08 2013 *)
PROG
(PARI) /* By Definition: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=Vec(sum(n=1, #A, (-x)^n*Ser(A)^(n%3)))[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Closed Form: */
{a(n)=local(A=1+x); polcoeff((1 - sqrt(4*(1-x)*(1+x^3) - 3 +x^2*O(x^n)))/(2*x), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 07 2013
STATUS
approved