A168055 Expansion of 2 - x - sqrt(1-2x-3x^2).

1, 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

Paul Barry, Nov 17 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A168054.

Examples

			G.f. = 1 + 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

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

Crossrefs

Cf. A168049.

Cf. A126068, A007971. [R. J. Mathar, Nov 18 2009]

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 2 - x - Sqrt[1 - 2 x - 3 x^2], {x, 0, n}] (* Michael Somos, Jan 25 2014 *)

PARI

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

Formula

a(n+2) = 2*A001006(n).

a(n) = 0^n + 2*Sum_{k=0..floor((n-2)/2)} C(n-2,2k)*A000108(k).

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

D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +3*(-n+3)*a(n-2)=0. - R. J. Mathar, Nov 19 2014

Extensions

Name corrected by Michael Somos, Mar 23 2012