A269956 Triangle read by rows, T(n,k) = binomial(3*n,n+k) for n>=0 and 0<=k<=n.

1, 3, 3, 15, 20, 15, 84, 126, 126, 84, 495, 792, 924, 792, 495, 3003, 5005, 6435, 6435, 5005, 3003, 18564, 31824, 43758, 48620, 43758, 31824, 18564, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 735471, 1307504, 1961256, 2496144, 2704156, 2496144, 1961256, 1307504, 735471

Offset: 0

Author

Peter Luschny, Mar 28 2016

Keywords

• nonn
• tabl

Examples

			Triangle starts:
1,
3,    3,
15,   20,   15,
84,   126,  126,  84,
495,  792,  924,  792,  495,
3003, 5005, 6435, 6435, 5005, 3003
		

Links

• Felix Fröhlich, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. binomial(m*n,n+k): A000007 (m=0), A010054 (m=1), A094527 (m=2).

Cf. A005809.

Programs

Magma

/* As triangle */ [[Binomial(3*n, n+k): k in [0..n]]: n in [0.. 7]]; // Vincenzo Librandi, Mar 28 2016

Maple

for n from 0 to 6 do seq(binomial(3*n,n+k), k=0..n) od;

Mathematica

Table[Binomial[3 n, n + k], {n, 0, 10}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Mar 28 2016 *)

PARI

t(n, k) = binomial(3*n, n+k)
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* The following function call prints the upper six rows of the triangle */
trianglerows(6) \\ Felix Fröhlich, Mar 28 2016

Formula

T(n,0) = T(n,n) = A005809(n).