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

%I A126068 #21 Oct 22 2017 23:35:22
%S A126068 0,0,2,2,4,8,18,42,102,254,646,1670,4376,11596,31022,83670,227268,
%T A126068 621144,1706934,4713558,13072764,36398568,101704038,285095118,
%U A126068 801526446,2259520830,6385455594,18086805002,51339636952,146015545604
%N A126068 Expansion of 1 - x - sqrt(1 - 2*x - 3*x^2) in powers of x.
%C A126068 Except for initial terms, identical to A007971.
%H A126068 Vincenzo Librandi, <a href="/A126068/b126068.txt">Table of n, a(n) for n = 0..1000</a>
%F A126068 G.f.: 1 - x - sqrt(1 - 2*x - 3*x^2). - _Michael Somos_, Jan 25 2014
%F A126068 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
%F A126068 a(n) ~ 3^(n-1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 20 2014
%e A126068 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 + ...
%p A126068 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);
%t A126068 a[ n_] := SeriesCoefficient[ 1 - x - Sqrt[1 - 2 x - 3 x^2], {x, 0, n}]; (* _Michael Somos_, Jan 25 2014 *)
%t A126068 CoefficientList[Series[1 - x - Sqrt[1 - 2 x - 3 x^2], {x, 0, 40}], x] (* _Vincenzo Librandi_, Apr 20 2014 *)
%o A126068 (PARI) {a(n) = polcoeff( (1 - x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n))), n)}; /* _Michael Somos_, Jan 25 2014 */
%Y A126068 Cf. A007971.
%K A126068 nonn
%O A126068 0,3
%A A126068 _Zerinvary Lajos_, Feb 28 2007
%E A126068 Better name by _Michael Somos_, Jan 25 2014