OFFSET
0,2
COMMENTS
By the application of enumerating Rota-Baxter word (not following the g.f.) the value at index 0 is set to a(0)=1.
Given y-2*y^2=x+x^2, expand y as a series in x, and then this sequence gives the coefficients: y=x+3*x^2+12*x^3+66*x^4+... (see PariGP code). - Robert Munafo, Oct 17 2024
LINKS
Robert Israel, Table of n, a(n) for n = 0..1057
L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337, theorem 3.6 at z=2.
FORMULA
D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +8*(-n+2)*a(n-2)=0. - R. J. Mathar, Jul 21 2017
a(n) ~ sqrt(3 - sqrt(6)) * 2^(n - 3/2) * (2 + sqrt(6))^(n+1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2024
MAPLE
f:= gfun:-rectoproc({8*n*a(n)+(12+8*n)*a(1+n)+(-3-n)*a(n+2), a(0) = 1, a(1) = 3}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jul 21 2017
MATHEMATICA
CoefficientList[Series[(1-Sqrt[1-8x-8x^2])/(4x), {x, 0, 30}], x] (* Harvey P. Dale, Feb 10 2018 *)
PROG
(PARI) my(x='x+O('x^99)); Vec((1-sqrt(1-8*x-8*x^2))/(4*x)) \\ Altug Alkan, Jul 22 2017
(PARI) my(y=x+O(x)); for(n=1, 23, y=x+x^2+2*y^2); Vec(y) \\ Robert Munafo, Oct 17 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jul 21 2017
STATUS
approved