OFFSET
1,5
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
FORMULA
T(n, k)=binomial(n, k+1)*binomial(n+2k-1, k)/(n+k) (0<=k<=n-1).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 3, 3;
1, 6, 14, 12;
1, 10, 40, 75, 55;
1, 15, 90, 275, 429, 273;
1, 21, 175, 770, 1911, 2548, 1428;
...
T(3,1)=3 because the noncrossing forests on 3 vertices A,B,C and having one edge are (A, BC), (B, CA) and (C, AB).
MAPLE
T:=proc(n, k) if k<=n-1 then binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k) else 0 fi end: seq(seq(T(n, k), k=0..n-1), n=1..11);
MATHEMATICA
T[n_, k_] := Binomial[n, k+1] Binomial[n+2k-1, k]/(n+k);
Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)
PROG
(PARI)
T(n, k)=binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k);
for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, May 31 2004
STATUS
approved