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
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.
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
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Feb 25 2018
STATUS
approved