A292440 Expansion of (1 - x + sqrt(1 - 2*x - 3*x^2))/2 in powers of x.

1, -1, -1, -1, -2, -4, -9, -21, -51, -127, -323, -835, -2188, -5798, -15511, -41835, -113634, -310572, -853467, -2356779, -6536382, -18199284, -50852019, -142547559, -400763223, -1129760415, -3192727797, -9043402501, -25669818476, -73007772802

Offset: 0

Author

Seiichi Manyama, Sep 16 2017

Keywords

• sign

Comments

Apart from a(1) the same as A168051. - R. J. Mathar, Sep 18 2017

Links

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

Crossrefs

Cf. A001006, A086246.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x +Sqrt(1-2*x-3*x^2))/2)); // G. C. Greubel, Aug 13 2018

Mathematica

CoefficientList[Series[(1-x +Sqrt[1-2*x-3*x^2])/2, {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *)

PARI

x='x+O('x^50); Vec((1 - x + sqrt(1 - 2*x - 3*x^2))/2) \\ G. C. Greubel, Aug 13 2018

Formula

Convolution inverse of A001006.

Let f(x) = (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2).

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

G.f.: 1/f(x) = 1 - x - x^2*f(x).

a(n) = -A001006(n-2) for n > 1.

a(n) ~ -3^(n - 1/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 14 2018

D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +3*(-n+3)*a(n-2)=0. - R. J. Mathar, Jan 23 2020