A321396 Square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 0, related to a class of Motzkin trees.

0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 3, 2, 5, 0, 1, 0, 1, 1, 3, 2, 7, 0, 0, 1, 0, 1, 1, 3, 3, 9, 5, 14, 0, 1, 0, 1, 1, 3, 3, 9, 7, 20, 0, 0, 1, 0, 1, 1, 3, 3, 10, 9, 27, 19, 42

Offset

0,21

Comments

The recursively specified combinatorial structure related to the array is the set of Motzkin trees where all leaves are at the same unary height (see section 3.2 in O. Bodini et al.).

Links

Table of n, a(n) for n=0..77.

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.

Formula

Define a sequence of generating functions recursively gf(-1) = 1 and for n >= 0

gf(n) = (1 - sqrt(1 - 4*z^2*gf(n-1)))/(2*z).

Row n of the array has the generating function gf(n)/z^n. For fixed k column k differs only for finitely many indices from the limit value A321397(k).

Example

Array begins:
    [0]  0, 1, 0, 1, 0, 2, 0,  5,  0, 14,  0,  42,   0, 132, ...  A126120
    [1]  0, 1, 0, 1, 1, 2, 2,  7,  5, 20, 19,  60,  62, 202, ...  A300126
    [2]  0, 1, 0, 1, 1, 3, 2,  9,  7, 27, 25,  85,  86, 287, ...  A321572
    [3]  0, 1, 0, 1, 1, 3, 3,  9,  9, 29, 32,  93, 111, 317, ...
    [4]  0, 1, 0, 1, 1, 3, 3, 10,  9, 31, 34, 100, 119, 344, ...
    [5]  0, 1, 0, 1, 1, 3, 3, 10, 10, 31, 36, 102, 126, 352, ...
    [6]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 36, 104, 128, 359, ...
    [7]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 104, 130, 361, ...
    [8]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 105, 130, 363, ...
    [9]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 105, 131, 363, ...
Array read by ascending diagonals:
    [0]  0
    [1]  0, 1
    [2]  0, 1, 0
    [3]  0, 1, 0, 1
    [4]  0, 1, 0, 1, 0
    [5]  0, 1, 0, 1, 1, 2
    [6]  0, 1, 0, 1, 1, 2, 0
    [7]  0, 1, 0, 1, 1, 3, 2, 5
    [8]  0, 1, 0, 1, 1, 3, 2, 7, 0
    [9]  0, 1, 0, 1, 1, 3, 3, 9, 5, 14

Maple

Arow := proc(n, len) local rowgf, ser;
rowgf := proc(n) option remember; `if`(n = 0, (1-sqrt(1-4*z^2))/(2*z),
expand((1 - sqrt(1 - 4*z^2*rowgf(n-1)))/(2*z))) end:
ser := series(rowgf(n)/z^n, z, 2*(2+max(len, n)));
seq(coeff(ser, z, k), k=0..len) end:
seq(Arow(n, 13), n=0..9);

Mathematica

nmax = 11; 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]^(nmax+1) // CoefficientList[#, z]&;
A[n_, k_] := row[n][[k + 1]];
Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 08 2018 *)

Crossrefs

Cf. A321395 (antidiagonal sums), A321397 (limit).

Cf. A000108 (Catalan), A001006 (Motzkin), A126120 (binary Catalan trees, row 0), A300126 (row 1), A321572 (row 2).

Sequence in context: A230000 A016242 A216659 * A141747 A239706 A328616

Adjacent sequences: A321393 A321394 A321395 * A321397 A321398 A321399

Keyword

nonn,tabl

Author

Peter Luschny, Nov 11 2018

Status

approved