A239425 Expansion of -16/(sqrt(12*x+2*sqrt(1-4*x)+2)-sqrt(1-4*x)-1)^2+1/x^2-1.

1, 2, 7, 16, 53, 156, 522, 1702, 5833, 19990, 70079, 247160, 882587, 3172196, 11492847, 41874864, 153452521, 564975570, 2089346157, 7756501690, 28898156364, 108010059036, 404890987653, 1521877280868, 5734545323859, 21657665796526

Offset: 0

Author

Vladimir Kruchinin, Mar 17 2014

Keywords

• nonn

Links

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

Crossrefs

Cf. A097609, A055113.

Programs

Mathematica

CoefficientList[Series[-16/(Sqrt[12*x+2*Sqrt[1-4*x]+2]-Sqrt[1-4*x] -1)^2+1/x^2-1, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 18 2014 *)
Flatten[{1,Table[Sum[Binomial[n+2*j-1,j+n-1]*(-1)^(j+n)*Binomial[2*n+2,j+n],{j,0,n+2}]/(n+1),{n,1,20}]}] (* Vaclav Kotesovec, Mar 18 2014 *)

Maxima

a(n):=(sum(binomial(n+2*j-1, j)*(-1)^(j+n)*binomial(2*n+2, j+n), j, 0, n+2))/(n+1)-kron_delta(n,0);

PARI

my(x='x+O('x^50)); Vec(-16/(sqrt(12*x+2*sqrt(1-4*x)+2)-sqrt(1-4*x) -1)^2 + 1/x^2 -1) \\ G. C. Greubel, Jun 01 2017

Formula

a(n) = (Sum_{j=0..(n+2)} C(n+2*j-1,j)*(-1)^(j+n)*C(2*n+2,j+n))/(n+1) - delta(n,0).

a(n) ~ (5+3*sqrt(5)) * 2^(2*n+1) / (5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 18 2014

Conjecture: 2*(2*n+1)*(n+2)*(n+1)*a(n) +(n+1)*(n^2-27*n+2)*a(n-1) +2*(-73*n^3+204*n^2-167*n+6)*a(n-2) +12*(n-3)*(2*n-3)*(4*n-7)*a(n-3) +216*(2*n-5)*(n-3)*(2*n-3)*a(n-4)=0. - R. J. Mathar, Apr 02 2014