A290147 Expansion of (1-sqrt(1-8*x-8*x^2))/(4*x).

1, 3, 12, 66, 408, 2712, 18912, 136488, 1010784, 7637664, 58650240, 456377664, 3590674176, 28516332288, 228297907200, 1840515987072, 14929020470784, 121749590032896, 997676696045568, 8210704762960896, 67835018440593408, 562407734010335232, 4677727530446635008

Offset: 0

Author

R. J. Mathar, Jul 21 2017

Keywords

• nonn

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.

Crossrefs

Cf. A025227.

Programs

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 *)

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

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