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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.

%I A321396 #23 Dec 08 2018 14:33:55
%S A321396 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,
%T A321396 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,
%U A321396 1,0,1,1,3,3,10,9,27,19,42
%N A321396 Square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 0, related to a class of Motzkin trees.
%C A321396 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.).
%H A321396 Olivier Bodini, Danièle Gardy, Bernhard Gittenberger, Zbigniew Gołębiewski, <a href="http://arxiv.org/abs/1510.01167">On the number of unary-binary tree-like structures with restrictions on the unary height</a>, arXiv:1510.01167v1 [math.CO], 2015.
%F A321396 Define a sequence of generating functions recursively gf(-1) = 1 and for n >= 0
%F A321396 gf(n) = (1 - sqrt(1 - 4*z^2*gf(n-1)))/(2*z).
%F A321396 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).
%e A321396 Array begins:
%e A321396     [0]  0, 1, 0, 1, 0, 2, 0,  5,  0, 14,  0,  42,   0, 132, ...  A126120
%e A321396     [1]  0, 1, 0, 1, 1, 2, 2,  7,  5, 20, 19,  60,  62, 202, ...  A300126
%e A321396     [2]  0, 1, 0, 1, 1, 3, 2,  9,  7, 27, 25,  85,  86, 287, ...  A321572
%e A321396     [3]  0, 1, 0, 1, 1, 3, 3,  9,  9, 29, 32,  93, 111, 317, ...
%e A321396     [4]  0, 1, 0, 1, 1, 3, 3, 10,  9, 31, 34, 100, 119, 344, ...
%e A321396     [5]  0, 1, 0, 1, 1, 3, 3, 10, 10, 31, 36, 102, 126, 352, ...
%e A321396     [6]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 36, 104, 128, 359, ...
%e A321396     [7]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 104, 130, 361, ...
%e A321396     [8]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 105, 130, 363, ...
%e A321396     [9]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 105, 131, 363, ...
%e A321396 Array read by ascending diagonals:
%e A321396     [0]  0
%e A321396     [1]  0, 1
%e A321396     [2]  0, 1, 0
%e A321396     [3]  0, 1, 0, 1
%e A321396     [4]  0, 1, 0, 1, 0
%e A321396     [5]  0, 1, 0, 1, 1, 2
%e A321396     [6]  0, 1, 0, 1, 1, 2, 0
%e A321396     [7]  0, 1, 0, 1, 1, 3, 2, 5
%e A321396     [8]  0, 1, 0, 1, 1, 3, 2, 7, 0
%e A321396     [9]  0, 1, 0, 1, 1, 3, 3, 9, 5, 14
%p A321396 Arow := proc(n, len) local rowgf, ser;
%p A321396 rowgf := proc(n) option remember; `if`(n = 0, (1-sqrt(1-4*z^2))/(2*z),
%p A321396 expand((1 - sqrt(1 - 4*z^2*rowgf(n-1)))/(2*z))) end:
%p A321396 ser := series(rowgf(n)/z^n, z, 2*(2+max(len, n)));
%p A321396 seq(coeff(ser, z, k), k=0..len) end:
%p A321396 seq(Arow(n, 13), n=0..9);
%t A321396 nmax = 11; gf[-1] = 1; gf[n_] := gf[n] = (1-Sqrt[1 - 4z^2 gf[n-1]])/(2z);
%t A321396 row[n_] := row[n] = gf[n]/z^n + O[z]^(nmax+1) // CoefficientList[#, z]&;
%t A321396 A[n_, k_] := row[n][[k + 1]];
%t A321396 Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 08 2018 *)
%Y A321396 Cf. A321395 (antidiagonal sums), A321397 (limit).
%Y A321396 Cf. A000108 (Catalan), A001006 (Motzkin), A126120 (binary Catalan trees, row 0), A300126 (row 1), A321572 (row 2).
%K A321396 nonn,tabl
%O A321396 0,21
%A A321396 _Peter Luschny_, Nov 11 2018