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

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

Author

Michael De Vlieger, Feb 25 2018

Keywords

• nonn

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Olivier Bodini, Paul Tarau, On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms, arXiv:1709.04302 [cs.PL], 2017.

Crossrefs

Cf. A000108, A001006, A135501, A321396 (row 1).

Programs

Magma

m:=50; R:=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

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 *)

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

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

Formula

G.f.: -(sqrt(2*z*(sqrt(1 - 4*z^2) - 1) + 1) - 1)/(2*z^2). - Peter Luschny, Nov 14 2018