A174117 Triangle T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 15, 40, 15, 1, 1, 24, 120, 120, 24, 1, 1, 35, 280, 525, 280, 35, 1, 1, 48, 560, 1680, 1680, 560, 48, 1, 1, 63, 1008, 4410, 7056, 4410, 1008, 63, 1, 1, 80, 1680, 10080, 23520, 23520, 10080, 1680, 80, 1, 1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1

Offset

0,5

Links

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

Formula

Let c(n) = Product_{j=2..n} (j^2 - 1) 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) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1.

T(n, k) = 2*((n+1)*(n-k)/(k+1))*A001263(n, k).

Sum_{k=0..n} T(n, k) = (2/(n+2))*( (n^2-1)*C_{n} + 1), where C_{n} are the Catalan numbers (A000108). (End)

Example

Triangle begins as:
  1;
  1,  1;
  1,  3,    1;
  1,  8,    8,     1;
  1, 15,   40,    15,     1;
  1, 24,  120,   120,    24,     1;
  1, 35,  280,   525,   280,    35,     1;
  1, 48,  560,  1680,  1680,   560,    48,     1;
  1, 63, 1008,  4410,  7056,  4410,  1008,    63,    1;
  1, 80, 1680, 10080, 23520, 23520, 10080,  1680,   80,  1;
  1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1;

Mathematica

(* First program *)
c[n_]:= If[n<2, 1, Product[i^2 -1, {i, 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, (2*k/(k+1))*Binomial[n+1, k]*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 (2*k/(k+1))*binomial(n-1, k)*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 (2*k/(k+1))*Binomial(n-1, k)*Binomial(n+1, k) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021

Crossrefs

Cf. A174116, A174119, A174124, A174125.

Cf. A000108, A001263.

Sequence in context: A094816 A097712 A238688 * A157210 A359573 A359575

Adjacent sequences: A174114 A174115 A174116 * A174118 A174119 A174120

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 08 2010

Extensions

Edited by G. C. Greubel, Feb 11 2021

Status

approved