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

%I A290147 #28 Oct 18 2024 18:47:12
%S A290147 1,3,12,66,408,2712,18912,136488,1010784,7637664,58650240,456377664,
%T A290147 3590674176,28516332288,228297907200,1840515987072,14929020470784,
%U A290147 121749590032896,997676696045568,8210704762960896,67835018440593408,562407734010335232,4677727530446635008
%N A290147 Expansion of (1-sqrt(1-8*x-8*x^2))/(4*x).
%C A290147 By the application of enumerating Rota-Baxter word (not following the g.f.) the value at index 0 is set to a(0)=1.
%C A290147 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
%H A290147 Robert Israel, <a href="/A290147/b290147.txt">Table of n, a(n) for n = 0..1057</a>
%H A290147 L. Guo and W. Y. Sit, <a href="http://dx.doi.org/10.1007/s11786-010-0061-2">Enumeration and generating functions of Rota-Baxter Words</a>, Math. Comput. Sci. 4 (2010) 313-337, theorem 3.6 at z=2.
%F A290147 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
%F A290147 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
%p A290147 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):
%p A290147 map(f, [$0..50]); # _Robert Israel_, Jul 21 2017
%t A290147 CoefficientList[Series[(1-Sqrt[1-8x-8x^2])/(4x),{x,0,30}],x] (* _Harvey P. Dale_, Feb 10 2018 *)
%o A290147 (PARI) my(x='x+O('x^99)); Vec((1-sqrt(1-8*x-8*x^2))/(4*x)) \\ _Altug Alkan_, Jul 22 2017
%o A290147 (PARI) my(y=x+O(x)); for(n=1,23,y=x+x^2+2*y^2); Vec(y) \\ _Robert Munafo_, Oct 17 2024
%Y A290147 Cf. A025227.
%K A290147 nonn
%O A290147 0,2
%A A290147 _R. J. Mathar_, Jul 21 2017