OFFSET
1,3
COMMENTS
Row n contains n terms. Column k=0 and row sums yield the ternary numbers (A001764).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
FORMULA
T(n, k) = ((k+1)(2k+1)/(3n-k-2)) binomial(3n-k-2, 2n-1).
G.f.: zg/(1-tzg^2)^2, where g = 1+zg^3 is the g.f. of the ternary numbers (A001764).
EXAMPLE
T(2,0)=1 and T(2,1)=2 because the noncrossing trees with 2 edges are /\, |_ and _|.
Triangle starts:
1;
1, 2;
3, 6, 3;
12, 24, 15, 4;
55, 110, 75, 28, 5;
273, 546, 390, 168, 45, 6;
...
MAPLE
T:=proc(n, k) if n=1 and k=1 then 0 elif k<=n then (k+1)*(2*k+1)*binomial(3*n-k-2, 2*n-1)/(3*n-k-2) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
MATHEMATICA
T[n_, k_] := ((k + 1)(2k + 1)/(3n - k - 2)) Binomial[3n - k - 2, 2n - 1];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Apr 10 2020 *)
PROG
(PARI)
T(n, k) = if(k<n, (k+1)*(2*k+1)*binomial(3*n-k-2, 2*n-1)/(3*n-k-2));
for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 06 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jan 15 2005
STATUS
approved