A321572 -id:A321572 - cp's OEIS frontend

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

Peter Luschny, Nov 11 2018

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.).

			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

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

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

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);

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 *)

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).

cp's OEIS Frontend

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

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