A182486 Expansion of 2 * (2 + x) / (4 - x - x * sqrt(1 - 4*x)) in powers of x.

1, 1, 0, -1, -2, -4, -10, -29, -90, -290, -960, -3246, -11164, -38934, -137358, -489341, -1757882, -6360634, -23160528, -84802606, -312041692, -1153271984, -4279311348, -15935808866, -59537435012, -223099337404, -838282693560, -3157706225584, -11922241414880

Offset: 0

Author

Michael Somos, May 02 2012

Keywords

• sign

Comments

HANKEL transform of sequence and the sequence omitting a(0) is the sequence A033999(n) = (-1)^n. This is the unique sequence with that property.

Examples

			G.f. = 1 + x - x^3 - 2*x^4 - 4*x^5 - 10*x^6 - 29*x^7 - 90*x^8 - 290*x^9 + ...
1 = det([ 1]) = det([ 1]). -1 = det([ 1, 1; 1, 0]) = det([ 1, 0; 0, -1]). 1 = det([ 1, 1, 0; 1, 0, -1; 0, -1, -2]) = det([ 1, 0, -1; 0, -1, -2; -2, -4, -10]).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

Crossrefs

Cf. A033999, A035929, A135334.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(2*(2+x)/(4-x -x*Sqrt(1-4*x)))); // G. C. Greubel, Aug 11 2018

Mathematica

CoefficientList[Series[2*(2+x)/(4-x -x*Sqrt[1-4*x]), {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

PARI

{a(n) = if( n<0, 0, polcoeff( 2 * (2 + x) / (4 - x - x * sqrt(1 - 4*x + x * O(x^n))) ,n))}

PARI

{a(n) = if( n<1, n==0, polcoeff( subst( (1 + x) / (1 + x^2), x, serreverse( x - x^2 + x * O(x^n))), n))}

Formula

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

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

G.f.: 1 / (1 - x / (1 + x / (1 - x / (1 - x / (1 - x / ...))))).

a(n) = -A135334(n - 2) if n > 2. a(n) - a(n+1) = A035929(n).

D-finite with recurrence: 2*(-n+1)*a(n) +2*(5*n-11)*a(n-1) +3*(-3*n+7)*a(n-2) +2*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jun 08 2016