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%I #35 Feb 26 2021 07:50:49
%S 1,0,-1,-1,-2,-4,-9,-21,-51,-127,-323,-835,-2188,-5798,-15511,-41835,
%T -113634,-310572,-853467,-2356779,-6536382,-18199284,-50852019,
%U -142547559,-400763223,-1129760415,-3192727797,-9043402501,-25669818476
%N Expansion of (1+x+sqrt(1-2x-3x^2))/2.
%C A signed variant of the Motzkin numbers A001006. Hankel transform is A168052.
%H G. C. Greubel, <a href="/A168051/b168051.txt">Table of n, a(n) for n = 0..1000</a>
%F D-finite with recurrence: n*a(n) -(2n-3)*a(n-1) -3*(n-3)*a(n-2)=0 if n>2. - _R. J. Mathar_, Dec 20 2011 [Edited by _Michael Somos_, Jan 25 2014]
%F 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 G.f.: 1 + x - (x + x^2) / (1 + x - (x + x^2) / (1 + x - ...)). - _Michael Somos_, Mar 27 2014
%F Convolution inverse of A005043. - _Michael Somos_, Mar 27 2014
%F a(n) ~ -3^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 05 2018
%F From _Gennady Eremin_, Feb 25 2021: (Start)
%F For n > 1, a(n) = A167022(n) / 2.
%F G.f.: (1 + x + A(x)) / 2, where A(x) is the g.f. of A167022. (End)
%e G.f. = 1 - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 21*x^7 - 51*x^8 - 127*x^9 + ...
%t a[ n_] := SeriesCoefficient[ (1 + x + Sqrt[1 - 2 x - 3 x^2]) / 2, {x, 0, n}] (* _Michael Somos_, Jan 25 2014 *)
%o (PARI) {a(n) = polcoeff( (1 + x + sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2, n)}; /* _Michael Somos_, Jan 25 2014 */
%Y Cf. A168049, A166587, A167022.
%Y Cf. A001006, A005043.
%K easy,sign
%O 0,5
%A _Paul Barry_, Nov 17 2009