OFFSET
0,2
COMMENTS
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
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, May 13 2006
STATUS
approved