A179648 Expansion of (1/(1+4x-2x^2))*c(x/(1+4x-2x^2)), c(x) the g.f. of A000108.

1, -3, 12, -47, 190, -778, 3224, -13475, 56710, -239986, 1020200, -4353430, 18636908, -80004388, 344264624, -1484499811, 6413133638, -27750688914, 120258432264, -521833284514, 2267084792708, -9859984425324, 42925569027408

Offset: 0

Author

Paul Barry, Jan 09 2011

Keywords

• sign
• easy

Comments

Hankel transform is the (4,5) Somos-4 sequence A174404.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1540

Crossrefs

Cf. A000108, A174404.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1/(2*x))*(1-Sqrt((1-2*x^2)/(1+4*x-2*x^2))))); // G. C. Greubel, Aug 14 2018

Mathematica

CoefficientList[Series[(1/(2*x))*(1 - Sqrt[(1-2*x^2)/(1+4*x-2*x^2)]), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)

PARI

x='x+O('x^50); Vec((1/(2*x))*(1-sqrt((1-2*x^2)/(1+4*x-2*x^2)))) \\ G. C. Greubel, Aug 14 2018

Formula

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

G.f.: 1/(1+4x-2x^2-x/(1-x/(1+4x-2x^2-x/(1-x/(1+4x-2x^2-x/(1-x/(1-... (continued fraction).

Conjecture: (n+1)*a(n) +2*(2n+1)*a(n-1) +4*(1-n)*a(n-2) +4*(5-2n)*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 17 2011

a(n) ~ (-1)^n * (2 + sqrt(6))^(n+1) / (2^(3/4) * 3^(1/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 15 2018