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A330800 Evaluation of the Motzkin polynomials at -1/2 and normalized with (-2)^n.

%I A330800 #8 Sep 14 2023 05:25:19
%S A330800 1,-1,5,-17,77,-345,1653,-8097,40733,-208553,1084421,-5708785,
%T A330800 30370861,-163019641,881790357,-4801746753,26302052925,-144825094473,
%U A330800 801155664933,-4450426297233,24815385947469,-138842668857369,779247587235765,-4385948395419873,24750623835149661
%N A330800 Evaluation of the Motzkin polynomials at -1/2 and normalized with (-2)^n.
%H A330800 G. C. Greubel, <a href="/A330800/b330800.txt">Table of n, a(n) for n = 0..1000</a>
%F A330800 a(n) = Sum_{k=0..n} (-1)^(n-k)*A201641(n,k).
%F A330800 a(n) = (-2)^n*Sum_{k=0..n} A064189(n,k)/(-2)^k.
%F A330800 a(n) = (-36*(n-2)*a(n-3) + 6*(4*n-5)*a(n-2) - (n-5)*a(n-1))/(n+1).
%F A330800 a(n) = [x^n] 2/(sqrt(4*x - 12*x^2 + 1) + 1).
%F A330800 a(n) = [x^n] reverse((x^2 + x)/(3*x^2 + 1))/x.
%p A330800 a := proc(n) option remember; if n < 3 then return [1, -1, 5][n+1] fi;
%p A330800 (-36*(n - 2)*a(n-3) + 6*(4*n - 5)*a(n-2) - (n - 5)*a(n-1))/(n + 1) end:
%p A330800 seq(a(n), n=0..24);
%p A330800 # Alternative:
%p A330800 gf := 2/(sqrt(4*x - 12*x^2 + 1) + 1):
%p A330800 ser := series(gf, x, 30): seq(coeff(ser,x,n), n=0..24);
%p A330800 # Or:
%p A330800 series((x^2+x)/(3*x^2+1), x, 30): gfun:-seriestoseries(%, 'revogf'):
%p A330800 convert(%, polynom) / x: seq(coeff(%, x, n), n=0..24);
%t A330800 A330800[n_]:= Coefficient[Series[2/(Sqrt[4*x-12*x^2+1] +1), {x,0,50}], x, n]; Table[A330800[n], {n, 0, 30}] (* _G. C. Greubel_, Sep 13 2023 *)
%o A330800 (SageMath)
%o A330800 R.<x> = PowerSeriesRing(QQ)
%o A330800 f = (x^2 + x)/(3*x^2 + 1)
%o A330800 f.reverse().shift(-1).list()
%o A330800 (Magma) I:=[1,-1,5]; [n le 3 select I[n] else ((6-n)*Self(n-1) + 6*(4*n-9)*Self(n-2) -36*(n-3)*Self(n-3))/n: n in [1..30]]; // _G. C. Greubel_, Sep 13 2023
%Y A330800 Cf. A064189, A129400, A201641, A330799.
%K A330800 sign
%O A330800 0,3
%A A330800 _Peter Luschny_, Jan 01 2020