A174116 Triangle T(n, k) = (n/2)*binomial(n-1, k-1)*binomial(n-1, k) with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 18, 6, 1, 1, 10, 60, 60, 10, 1, 1, 15, 150, 300, 150, 15, 1, 1, 21, 315, 1050, 1050, 315, 21, 1, 1, 28, 588, 2940, 4900, 2940, 588, 28, 1, 1, 36, 1008, 7056, 17640, 17640, 7056, 1008, 36, 1, 1, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 1

Offset: 0

Author

Roger L. Bagula, Mar 08 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,    1;
  1,  3,    3,     1;
  1,  6,   18,     6,     1;
  1, 10,   60,    60,    10,     1;
  1, 15,  150,   300,   150,    15,     1;
  1, 21,  315,  1050,  1050,   315,    21,     1;
  1, 28,  588,  2940,  4900,  2940,   588,    28,    1;
  1, 36, 1008,  7056, 17640, 17640,  7056,  1008,   36,  1;
  1, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A174117, A174119, A174124, A174125.

Cf. A000108, A001263.

Programs

Magma

T:= func< n,k | k eq 0 or k eq n select 1 else (n/2)*Binomial(n-1, k-1)*Binomial(n-1, k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021

Mathematica

(* First program *)
c[n_]:= If[n<2, 1, Product[Binomial[j,2], {j, 2, n}]];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, (n/2)*Binomial[n-1, k-1]*Binomial[n-1, k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)

Sage

def T(n,k): return 1 if (k==0 or k==n) else (n/2)*binomial(n-1, k-1)*binomial(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

Formula

Let c(n) = Product_{j=2..n} binomial(j,2) for n > 1 otherwise 1 then the number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).

From G. C. Greubel, Feb 11 2021: (Start)

T(n, k) = (n/2)*binomial(n-1, k-1)*binomial(n-1, k) with T(n, 0) = T(n, n) = 1.

T(n, k) = binomial(n-k+1, 2)*A001263(n, k) with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n,k) = binomial(n, 2)*C_{n-1} + 2 - [n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. (End)

Extensions

Edited by G. C. Greubel, Feb 11 2021