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A321396
Square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 0, related to a class of Motzkin trees.
4
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
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 11 2018
STATUS
approved