A189230 Complementary Catalan triangle read by rows.

0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 0, 8, 0, 4, 0, 10, 0, 15, 0, 5, 0, 0, 30, 0, 24, 0, 6, 0, 35, 0, 63, 0, 35, 0, 7, 0, 0, 112, 0, 112, 0, 48, 0, 8, 0, 126, 0, 252, 0, 180, 0, 63, 0, 9, 0, 0, 420, 0, 480, 0, 270, 0, 80, 0, 10, 0, 462, 0, 990, 0, 825, 0, 385, 0, 99, 0, 11, 0

Offset

0,5

Comments

T(n,k) = A189231(n,k)*((n - k) mod 2). For comparison: the classical Catalan triangle is A053121(n,k) = A189231(n,k)*((n-k+1) mod 2).

T(n,0) = A138364(n). Row sums: A100071.

Links

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

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Peter Luschny, The lost Catalan numbers

Example

[0]  0,
[1]  1,  0,
[2]  0,  2,  0,
[3]  3,  0,  3,  0,
[4]  0,  8,  0,  4,  0,
[5] 10,  0, 15,  0,  5, 0,
[6]  0, 30,  0, 24,  0, 6, 0,
[7] 35,  0, 63,  0, 35, 0, 7, 0,
   [0],[1],[2],[3],[4],[5],[6],[7]

Maple

A189230 := (n, k) -> A189231(n, k)*modp(n-k, 2):
seq(print(seq(A189230(n, k), k=0..n)), n=0..11);

Mathematica

t[n_, k_] /; (k>n || k<0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2] t[n-1, k] + t[n-1, k+1];
T[n_, k_] := t[n, k] Mod[n-k, 2];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jun 24 2019 *)

Crossrefs

Cf. A053121, A162246, A057977, A189231.

Sequence in context: A011374 A298645 A243319 * A243982 A214000 A161123

Adjacent sequences: A189227 A189228 A189229 * A189231 A189232 A189233

Keyword

nonn,tabl

Author

Peter Luschny, May 01 2011

Status

approved