A119301 Triangle read by rows: T(n,k) = binomial(3*n-k,n-k).

1, 3, 1, 15, 5, 1, 84, 28, 7, 1, 495, 165, 45, 9, 1, 3003, 1001, 286, 66, 11, 1, 18564, 6188, 1820, 455, 91, 13, 1, 116280, 38760, 11628, 3060, 680, 120, 15, 1, 735471, 245157, 74613, 20349, 4845, 969, 153, 17, 1, 4686825, 1562275, 480700, 134596, 33649, 7315

Offset: 0

Author

Paul Barry, May 13 2006

Keywords

• easy
• nonn
• tabl

Comments

First column is A005809. Second column is A025174.

Row sums are A045721. Inverse is Riordan array (1-3x,x(1-x)^2), A119302.

Examples

			Triangle begins
  1,
  3, 1,
  15, 5, 1,
  84, 28, 7, 1,
  495, 165, 45, 9, 1,
  3003, 1001, 286, 66, 11, 1,
  18564, 6188, 1820, 455, 91, 13, 1,
  116280, 38760, 11628, 3060, 680, 120, 15, 1
  ...
Horizontal recurrence: T(4,1) = 1*84 + 2*28 + 3*7 + 4*1 = 165. - _Peter Bala_, Dec 29 2014
		

Crossrefs

Cf. A092392, A119304.

Cf. A005809, A025174, A045721, A119302.

Programs

Maple

T := proc(n,k) option remember;
`if`(n = 0, 1, add(i*T(n-1,k-2+i),i=1..n+1-k)) end:
for n from 0 to 9 do print(seq(T(n,k),k=0..n)) od; # Peter Luschny, Dec 30 2014

Mathematica

Flatten[Table[Binomial[3n-k,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 28 2012 *)

Formula

G.f. g(x) = 2*sin(arcsin(3*sqrt(3*x)/2)/3)/sqrt(3*x) satisfies g(x) = 1/(1-x*g(x)^2).

Riordan array (1/(1-3*x*g(x)^2),x*g(x)^2) where g(x)=1+x*g(x)^3.

'Horizontal' recurrence equation: T(n,0) = binomial(3*n,n) and for k >= 1, T(n,k) = Sum_{i = 1..n+1-k} i*T(n-1,k-2+i). - Peter Bala, Dec 28 2014

T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(2*n-k-j, n). - Peter Bala, Jun 04 2024