A321396 -id:A321396 - cp's OEIS frontend

A321397 a is the limit of A321396.

0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 105, 131, 364, 464, 1269, 1690, 4478, 6163, 16070, 22582, 58197, 83431, 212337, 309890, 780706, 1155906, 2888189, 4331109, 10739640, 16291293, 40123020, 61483211, 150518418, 232754753, 566704138, 883597884, 2140614523

Offset: 0

Peter Luschny, Nov 11 2018

nonn

The sequence a is the limit of the square array A321396 means, that for every fixed k the terms in column k of A321396 differ from a(k) only for finitely many indices.

Cf. A321396.

terms = 38; gf[-1] = 1; gf[n_] := gf[n] = (1-Sqrt[1-4z^2 gf[n -1]])/(2z);
row[n_] := row[n] = gf[n]/z^n + O[z]^terms // CoefficientList[#, z]&;
row[n = 1]; row[n++]; While[Print[n]; row[n] != row[n-1], n++];
A321397 = row[n] (* Jean-François Alcover, Dec 08 2018 *)

A321395 Antidiagonal sums of A321396.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 5, 13, 15, 37, 45, 116, 142, 371, 475, 1226, 1610, 4175, 5561, 14454, 19574, 50743, 69725, 180503, 250777, 648541, 910317, 2349582, 3329020, 8575583, 12250279, 31498125, 45331627

Offset: 0

Peter Luschny, Nov 11 2018

nonn

Cf. A321396.

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

Original entry on oeis.org

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

Michael De Vlieger, Feb 25 2018

nonn

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

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.

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

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

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

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 *)

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

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

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

A321572 Related to the set of Motzkin trees where all leaves are at the same unary height 2.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 2, 9, 7, 27, 25, 85, 86, 287, 296, 975, 1065, 3369, 3825, 11887, 13836, 42389, 50597, 152549, 186186, 554103, 688494, 2027304, 2559958, 7461971, 9561298, 27617581, 35846863, 102707431, 134874639, 383561963, 509090498, 1437822479, 1927045425

Offset: 0

Peter Luschny, Nov 14 2018

nonn

Row 2 of A321396, see section 3.2 in O. Bodini et al.

Olivier Bodini, Danièle Gardy, Bernhard Gittenberger, Zbigniew Gołębiewski, On the number of unary-binary tree-like structures with restrictions on the unary height, arXiv:1510.01167v1 [math.CO], 2015.

Cf. A321396.

gf := -(sqrt(2*z*(sqrt(2*z*(sqrt(1-4*z^2)-1)+1)-1)+1)-1)/(2*z^3):
series(gf,z,44): seq(coeff(%,z,n), n=0..38);

CoefficientList[(1 - Sqrt[2 Sqrt[2 Sqrt[1 - 4z^2] z - 2z + 1] z - 2z + 1])/ (2z^3) + O[z]^40, z] (* Jean-François Alcover, Jun 03 2019 *)

G.f.: (1 - sqrt(1 - 2*z + 2*z*sqrt(1 - 2*z + 2*z*sqrt(1 - 4*z^2))))/(2*z^3).

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A321397 a is the limit of A321396.

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A321395 Antidiagonal sums of A321396.

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

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A321572 Related to the set of Motzkin trees where all leaves are at the same unary height 2.

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