A174119 Triangle T(n, k) = ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k) with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 14, 70, 14, 1, 1, 30, 420, 420, 30, 1, 1, 55, 1650, 4620, 1650, 55, 1, 1, 91, 5005, 30030, 30030, 5005, 91, 1, 1, 140, 12740, 140140, 300300, 140140, 12740, 140, 1, 1, 204, 28560, 519792, 2042040, 2042040, 519792, 28560, 204, 1, 1, 285, 58140, 1627920, 10581480, 19399380, 10581480, 1627920, 58140, 285, 1

Offset

0,8

Links

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

Formula

T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} j*(j-1)*(2*j-1)/6 for n > 2 otherwise 1.

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

T(n, k) = ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k) with T(n, 0) = T(n, n) =1.

Sum_{k=0..n} T(n, k) = (n*(n-1)*(2*n-1)/6)*HypergeometricPFQ[{1-n, 3/2-n, 2-n}, {3/2, 2}, -1] + 2 - [n=0] (n*(n-1)*(2*n-1)/6)*A196148[n-2] + 2 - [n=0]. (End)

Example

Triangle begins as:
  1;
  1,   1;
  1,   1,     1;
  1,   5,     5,       1;
  1,  14,    70,      14,        1;
  1,  30,   420,     420,       30,        1;
  1,  55,  1650,    4620,     1650,       55,        1;
  1,  91,  5005,   30030,    30030,     5005,       91,       1;
  1, 140, 12740,  140140,   300300,   140140,    12740,     140,     1;
  1, 204, 28560,  519792,  2042040,  2042040,   519792,   28560,   204,   1;
  1, 285, 58140, 1627920, 10581480, 19399380, 10581480, 1627920, 58140, 285, 1;

Mathematica

(* First program *)
c[n_]:= If[n<2, 1, Product[j*(j-1)*(2*j-1)/6, {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-k)/6)*Binomial[n-1, k-1]*Binomial[2*n, 2*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-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*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-k)/6)*Binomial(n-1, k-1)*Binomial(2*n, 2*k) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021

Crossrefs

Cf. A174116, A174117, A174124, A174125.

Cf. A111910, A196148.

Sequence in context: A172349 A144403 A188587 * A156696 A232651 A145227

Adjacent sequences: A174116 A174117 A174118 * A174120 A174121 A174122

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 08 2010

Extensions

Edited by G. C. Greubel, Feb 11 2021

Status

approved