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A300126
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Number of Motzkin trees that are "uniquely closable skeletons".
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2
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0, 1, 0, 1, 1, 2, 2, 7, 5, 20, 19, 60, 62, 202, 202, 679, 711, 2304, 2507, 8046, 8856, 28434, 31855, 101288, 115596, 364710, 421654, 1323946, 1549090, 4836072, 5724582, 17771683, 21250527, 65653884, 79227989, 243639954, 296543356, 907841678, 1113706887
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OFFSET
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0,6
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COMMENTS
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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".
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
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LINKS
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FORMULA
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G.f.: -(sqrt(2*z*(sqrt(1 - 4*z^2) - 1) + 1) - 1)/(2*z^2). - Peter Luschny, Nov 14 2018
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MAPLE
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gf := -(sqrt(2*z*(sqrt(1 - 4*z^2) - 1) + 1) - 1)/(2*z^2):
series(gf, z, 44): seq(coeff(%, z, n), n=0..38); # Peter Luschny, Nov 14 2018
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MATHEMATICA
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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 *)
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PROG
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(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
(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
(Sage) s= (-(sqrt(2*x*(sqrt(1 - 4*x^2) - 1) + 1) - 1)/(2*x^2)).series(x, 30);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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