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Complementary Catalan triangle read by rows.
3

%I #12 May 08 2020 17:45:33

%S 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,

%T 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,

%U 480,0,270,0,80,0,10,0,462,0,990,0,825,0,385,0,99,0,11,0

%N Complementary Catalan triangle read by rows.

%C 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).

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

%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>

%e [0] 0,

%e [1] 1, 0,

%e [2] 0, 2, 0,

%e [3] 3, 0, 3, 0,

%e [4] 0, 8, 0, 4, 0,

%e [5] 10, 0, 15, 0, 5, 0,

%e [6] 0, 30, 0, 24, 0, 6, 0,

%e [7] 35, 0, 63, 0, 35, 0, 7, 0,

%e [0],[1],[2],[3],[4],[5],[6],[7]

%p A189230 := (n,k) -> A189231(n,k)*modp(n-k,2):

%p seq(print(seq(A189230(n,k),k=0..n)),n=0..11);

%t 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 T[n_, k_] := t[n, k] Mod[n-k, 2];

%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* _Jean-François Alcover_, Jun 24 2019 *)

%Y Cf. A053121, A162246, A057977, A189231.

%K nonn,tabl

%O 0,5

%A Peter Luschny, May 01 2011