A173993 Sequence whose Hankel transform is the Somos (4) sequence.

1, 2, 6, 17, 50, 146, 430, 1267, 3746, 11091, 32900, 97716, 290586, 864980, 2577032, 7683397, 22922874, 68427057, 204362172, 610604629, 1825092080, 5457016431, 16321318264, 48828168580, 146112907266, 437319580738, 1309158060068

Offset: 0

Author

Paul Barry, Mar 04 2010

Keywords

• easy
• nonn

Comments

Hankel transform is A006720(n+3).

Links

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

Crossrefs

Cf. A006720, A173992.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((Sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3))-2*x^2+4*x-1)/(2*x*(1-4*x+3*x^2)))); // G. C. Greubel, Sep 22 2018

Mathematica

CoefficientList[Series[(Sqrt[(1-2*x)*(1-2*x-4*x^2+4*x^3)]-2*x^2+4*x-1)/( 2 x*(1 - 4 x + 3 x^2)), {x, 0, 50}], x] (* G. C. Greubel, Sep 22 2018 *)

PARI

my(x='x+O('x^50)); Vec((sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3))-2*x^2+4*x-1)/(2*x*(1-4*x+3*x^2))) \\ G. C. Greubel, Sep 22 2018

Formula

G.f.: (sqrt((1-2x)*(1-2x-4x^2+4x^3))-2x^2+4x-1)/(2x*(1-4x+3x^2)).

Conjecture: (n+1)*a(n) +2*(-4*n-1)*a(n-1) +(19*n-5)*a(n-2) -36*a(n-3) +8*(-7*n+26)*a(n-4) +2*(34*n-143)*a(n-5) +24*(-n+5)*a(n-6)=0. - R. J. Mathar, Oct 10 2014