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

%I A168055 #22 Jan 30 2020 21:29:16
%S A168055 1,0,2,2,4,8,18,42,102,254,646,1670,4376,11596,31022,83670,227268,
%T A168055 621144,1706934,4713558,13072764,36398568,101704038,285095118,
%U A168055 801526446,2259520830,6385455594,18086805002,51339636952,146015545604
%N A168055 Expansion of 2 - x - sqrt(1-2x-3x^2).
%C A168055 Hankel transform is A168054.
%H A168055 G. C. Greubel, <a href="/A168055/b168055.txt">Table of n, a(n) for n = 0..1000</a>
%F A168055 a(n+2) = 2*A001006(n).
%F A168055 a(n) = 0^n + 2*Sum_{k=0..floor((n-2)/2)} C(n-2,2k)*A000108(k).
%F A168055 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 A168055 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
%e A168055 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 + ...
%t A168055 a[ n_] := SeriesCoefficient[ 2 - x - Sqrt[1 - 2 x - 3 x^2], {x, 0, n}] (* _Michael Somos_, Jan 25 2014 *)
%o A168055 (PARI) {a(n) = polcoeff( 2 - x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n)), n)} /* _Michael Somos_, Jan 25 2014 */
%Y A168055 Cf. A168049.
%Y A168055 Cf. A126068, A007971. [_R. J. Mathar_, Nov 18 2009]
%K A168055 easy,nonn
%O A168055 0,3
%A A168055 _Paul Barry_, Nov 17 2009
%E A168055 Name corrected by _Michael Somos_, Mar 23 2012