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

0,3

Comments

Except for initial terms, identical to A007971.

Links

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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A007971.

Sequence in context: A139800 A168058 A007971 * A167022 A168055 A005702

Adjacent sequences: A126065 A126066 A126067 * A126069 A126070 A126071

Keyword

nonn

Author

Zerinvary Lajos, Feb 28 2007

Extensions

Better name by Michael Somos, Jan 25 2014

Status

approved