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A300126 Number of Motzkin trees that are "uniquely closable skeletons".

%I A300126 #16 Sep 08 2022 08:46:20
%S A300126 0,1,0,1,1,2,2,7,5,20,19,60,62,202,202,679,711,2304,2507,8046,8856,
%T A300126 28434,31855,101288,115596,364710,421654,1323946,1549090,4836072,
%U A300126 5724582,17771683,21250527,65653884,79227989,243639954,296543356,907841678,1113706887
%N A300126 Number of Motzkin trees that are "uniquely closable skeletons".
%C A300126 From the Bodini-Tarau paper: "Uniquely closable skeletons of lambda terms are Motzkin-trees that predetermine the unique closed lambda term that can be obtained by labeling their leaves with de Bruijn indices".
%C A300126 For the relation to the set of Motzkin trees where all leaves are at the same unary height see A321396. - _Peter Luschny_, Nov 14 2018
%H A300126 G. C. Greubel, <a href="/A300126/b300126.txt">Table of n, a(n) for n = 0..1000</a>
%H A300126 Olivier Bodini, Paul Tarau, <a href="https://arxiv.org/abs/1709.04302">On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms</a>, arXiv:1709.04302 [cs.PL], 2017.
%F A300126 G.f.: -(sqrt(2*z*(sqrt(1 - 4*z^2) - 1) + 1) - 1)/(2*z^2). - _Peter Luschny_, Nov 14 2018
%p A300126 gf := -(sqrt(2*z*(sqrt(1 - 4*z^2) - 1) + 1) - 1)/(2*z^2):
%p A300126 series(gf, z, 44): seq(coeff(%, z, n), n=0..38); # _Peter Luschny_, Nov 14 2018
%t A300126 CoefficientList[Series[(1-Sqrt[1 + 2*x*(Sqrt[1-4*x^2]-1)])/(2*x^2), {x,0, 50}], x] (* _G. C. Greubel_, Nov 14 2018 *)
%o A300126 (PARI) x='x+O('x^50); concat([0], Vec((1-sqrt(1 + 2*x*(sqrt(1-4*x^2) -1)))/(2*x^2))) \\ _G. C. Greubel_, Nov 14 2018
%o A300126 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( (1-Sqrt(1 + 2*x*(Sqrt(1-4*x^2) -1)))/(2*x^2) )); // _G. C. Greubel_, Nov 14 2018
%o A300126 (Sage) s= (-(sqrt(2*x*(sqrt(1 - 4*x^2) - 1) + 1) - 1)/(2*x^2)).series(x, 30);
%o A300126 s.coefficients(x, sparse=False) # _G. C. Greubel_, Nov 14 2018
%Y A300126 Cf. A000108, A001006, A135501, A321396 (row 1).
%K A300126 nonn
%O A300126 0,6
%A A300126 _Michael De Vlieger_, Feb 25 2018