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

Author

Peter Luschny, Nov 11 2018

Keywords

• nonn
• tabl

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.).

Examples

			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
		

Links

• 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.

Crossrefs

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

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

Programs

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 *)

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).