%I #20 Apr 26 2024 09:05:14

%S 1,1,1,1,4,1,1,7,10,1,1,10,28,20,1,1,13,55,84,35,1,1,16,91,220,210,56,

%T 1,1,19,136,455,715,462,84,1,1,22,190,816,1820,2002,924,120,1,1,25,

%U 253,1330,3876,6188,5005,1716,165,1,1,28,325,2024,7315,15504,18564

%N Triangle: T(n,k) = C(3n-2k,k), 0 <= k <= n.

%H Robert Israel, <a href="/A193636/b193636.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)

%F T(n,k) = C(3n-2k,k), 0 <= k <= n.

%F G.f. as triangle: (1-x*y)^2/(1 - x - 3*x*y + 3*x^2*y^2 - x^3*y^3). - _Robert Israel_, Nov 06 2018

%F T(n,k) = A102547(3*n,k). - _R. J. Mathar_, Apr 26 2024

%e First 5 rows:

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 7, 10, 1;

%e 1, 10, 28, 20, 1; [Corrected by _Robert Israel_, Nov 06 2018]

%p seq(seq(binomial(3*n-2*k,k),k=0..n),n=0..10); # _Robert Israel_, Nov 06 2018

%t p[n_, k_] := Binomial[3 n - 2 k, k];

%t Table[p[n, k], {n, 0, 9}, {k, 0, n}] (* A193636 *)

%t Flatten[%]

%o (Magma) /* As triangle */[[Binomial(3*n-2*k,k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Nov 07 2018

%Y Cf. A193635.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 01 2011