A271197 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.

1, 7, 43, 255, 1491, 8659, 50107, 289375, 1669291, 9623427, 55460835, 319583823, 1841526051, 10612066227, 61160827083, 352543298751, 2032480521819, 11719811413027, 67592446165363, 389906344880815, 2249609496664531

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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))).

a(n) ~ (3+2*sqrt(2))^(n+2) / (2^(5/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Apr 01 2016

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

Mathematica

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 *)
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 *)

Prog

(Maxima)
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);
(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

Crossrefs

Cf. A001003.

Sequence in context: A049609 A161728 A358682 * A240366 A329018 A003464

Adjacent sequences: A271194 A271195 A271196 * A271198 A271199 A271200

Keyword

nonn,easy

Author

Vladimir Kruchinin, Apr 01 2016

Status

approved