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A300126
Number of Motzkin trees that are "uniquely closable skeletons".
2
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
OFFSET
0,6
COMMENTS
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
LINKS
Olivier Bodini, Paul Tarau, On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms, arXiv:1709.04302 [cs.PL], 2017.
FORMULA
G.f.: -(sqrt(2*z*(sqrt(1 - 4*z^2) - 1) + 1) - 1)/(2*z^2). - Peter Luschny, Nov 14 2018
MAPLE
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
MATHEMATICA
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 *)
PROG
(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);
s.coefficients(x, sparse=False) # G. C. Greubel, Nov 14 2018
CROSSREFS
Cf. A000108, A001006, A135501, A321396 (row 1).
Sequence in context: A249493 A223000 A058625 * A006748 A193548 A131049
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Feb 25 2018
STATUS
approved