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

0,8

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A174117, A174119, A174124, A174125.

Cf. A000108, A001263.

Sequence in context: A083029 A084546 A288266 * A270273 A026515 A075772

Adjacent sequences: A174113 A174114 A174115 * A174117 A174118 A174119

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 08 2010

Extensions

Edited by G. C. Greubel, Feb 11 2021

Status

approved