A145890 Triangle read by rows: T(n,k) = B(k)C(n-k), where B(j) is the central binomial coefficient binomial(2j,j) (A000984) and C(j) is the Catalan number binomial(2j,j)/(j+1) (A000108); 0 <= k <= n.

1, 1, 2, 2, 2, 6, 5, 4, 6, 20, 14, 10, 12, 20, 70, 42, 28, 30, 40, 70, 252, 132, 84, 84, 100, 140, 252, 924, 429, 264, 252, 280, 350, 504, 924, 3432, 1430, 858, 792, 840, 980, 1260, 1848, 3432, 12870, 4862, 2860, 2574, 2640, 2940, 3528, 4620, 6864, 12870, 48620

Offset: 0

Author

Emeric Deutsch, Nov 22 2008

Keywords

• nonn
• tabl

Comments

Sum of entries in row n is (1/2)binomial(2n+2,n+1) = A001700(n).

T(n,0) = binomial(2n,n)/(n+1) = A000108(n) (the Catalan numbers).

T(n,n) = binomial(2n,n) = A000984(n) (the central binomial coefficients).

Examples

			Triangle starts:
   1;
   1,  2;
   2,  2,  6;
   5,  4,  6, 20;
  14, 10, 12, 20, 70;
		

Crossrefs

Cf. A000108, A000984, A001700.

Programs

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 *)

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)).