OFFSET
0,3
COMMENTS
FORMULA
T(n,k) = binomial(2k,k)*binomial(2n-2k,n-k)/(n-k+1) (0 <= k <= n).
G.f. = G(t,x) = (1-sqrt(1-4x))/(2x*sqrt(1-4tx)).
EXAMPLE
Triangle starts:
1;
1, 2;
2, 2, 6;
5, 4, 6, 20;
14, 10, 12, 20, 70;
MAPLE
b:=proc(n) options operator, arrow: binomial(2*n, n) end proc: c:=proc(n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T:=proc(n, k) if k <= n then b(k)*c(n-k) else 0 end if end proc: for n from 0 to 8 do seq(T(n, k), k =0..n) end do; # yields sequence in triangular form
MATHEMATICA
T[n_, k_] /; 0 <= k <= n := Binomial[2k, k]*Binomial[2n - 2k, n-k]/(n-k+1);
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 23 2024 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Nov 22 2008
STATUS
approved