A126068 Expansion of 1 - x - sqrt(1 - 2*x - 3*x^2) in powers of x.

0, 0, 2, 2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604

Offset: 0

Author

Zerinvary Lajos, Feb 28 2007

Keywords

• nonn

Comments

Except for initial terms, identical to A007971.

Examples

			G.f. = 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 42*x^7 + 102*x^8 + 254*x^9 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A007971.

Programs

Maple

zl:=4*(1-z+sqrt(1-2*z-3*z^2))/(1-z+sqrt(1-2*z-3*z^2))^2: gser:=series(zl, z=0, 35): seq(coeff(gser, z, n), n=-2..27);

Mathematica

a[ n_] := SeriesCoefficient[ 1 - x - Sqrt[1 - 2 x - 3 x^2], {x, 0, n}]; (* Michael Somos, Jan 25 2014 *)
CoefficientList[Series[1 - x - Sqrt[1 - 2 x - 3 x^2], {x, 0, 40}], x] (* Vincenzo Librandi, Apr 20 2014 *)

PARI

{a(n) = polcoeff( (1 - x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n))), n)}; /* Michael Somos, Jan 25 2014 */

Formula

G.f.: 1 - x - sqrt(1 - 2*x - 3*x^2). - Michael Somos, Jan 25 2014

0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 25 2014

a(n) ~ 3^(n-1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 20 2014

Extensions

Better name by Michael Somos, Jan 25 2014