%I #17 Jan 30 2020 21:29:17
%S 1,7,43,255,1491,8659,50107,289375,1669291,9623427,55460835,319583823,
%T 1841526051,10612066227,61160827083,352543298751,2032480521819,
%U 11719811413027,67592446165363,389906344880815,2249609496664531
%N Expansion of -(sqrt(x^2-6*x+1)+3*x-1)/((7*x-1)*sqrt(x^2-6*x+1)+x^2-6*x+1)/x.
%H G. C. Greubel, <a href="/A271197/b271197.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n}(Sum_{j=0..n+1}(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j))).
%F a(n) ~ (3+2*sqrt(2))^(n+2) / (2^(5/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 01 2016
%F D-finite with recurrence: (n+1)*a(n) +(-17*n-4)*a(n-1) +(97*n-57)*a(n-2) +(-191*n+280)*a(n-3) +30*(n-2)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020
%t CoefficientList[Series[-(Sqrt[x^2 - 6 x + 1] + 3 x - 1)/((7 x - 1) Sqrt[x^2 - 6 x + 1] + x^2 - 6 x + 1)/x, {x, 0, 20}], x] (* or *)
%t Table[Sum[Sum[Binomial[j, n - k - j] 3^(-n + k + 2 j) 2^(n - k - j) Binomial[n + 1, j], {j, 0, n + 1}], {k, 0, n}], {n, 0, 20}] (* _Michael De Vlieger_, Apr 01 2016 *)
%o (Maxima)
%o a(n):=sum(sum(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j),j,0,n+1),k,0,n);
%o (PARI) for(n=0,25, print1(sum(k=0,n,(sum(j=0,n+1,(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j))))), ", ")) \\ _G. C. Greubel_, Jun 05 2017
%Y Cf. A001003.
%K nonn,easy
%O 0,2
%A _Vladimir Kruchinin_, Apr 01 2016