A157243 Triangle T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 336, 10, 1, 1, 15, 825, 825, 15, 1, 1, 21, 1716, 197676, 1716, 21, 1, 1, 28, 3185, 512050, 512050, 3185, 28, 1, 1, 36, 5440, 1163800, 294296640, 1163800, 5440, 36, 1, 1, 45, 8721, 2395575, 778076145, 778076145, 2395575, 8721, 45, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k.

T(n, n-k) = T(n, k).

T(n, 1) = A000217(n). - G. C. Greubel, Jan 11 2022

Example

Triangle begins as:
  1;
  1,  1;
  1,  3,    1;
  1,  6,    6,       1;
  1, 10,  336,      10,         1;
  1, 15,  825,     825,        15,         1;
  1, 21, 1716,  197676,      1716,        21,       1;
  1, 28, 3185,  512050,    512050,      3185,      28,    1;
  1, 36, 5440, 1163800, 294296640,   1163800,    5440,   36,  1;
  1, 45, 8721, 2395575, 778076145, 778076145, 2395575, 8721, 45, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
A001263[n_, k_]:= Binomial[n-1, k-1]*Binomial[n, k]/(n-k+1);
T[n_, k_]:= A001263[n*f[n, k] +1, f[n, k] +1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)

Prog

(Magma)
f:= func< n, k | k le Floor(n/2) select k else n-k >;
A001263:= func< n, k | Binomial(n-1, k-1)*Binomial(n, k)/(n-k+1) >;
[A001263(n*f(n, k)+1, f(n, k)+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022
(Sage)
def f(n, k): return k if (k <= (n//2)) else n-k
def A001263(n, k): return binomial(n-1, k-1)*binomial(n, k)/(n-k+1)
flatten([[A001263(n*f(n, k)+1, f(n, k)+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022

Crossrefs

Cf. A000217, A001263, A157219, A157221.

Sequence in context: A180959 A131235 A202812 * A146769 A189610 A172427

Adjacent sequences: A157240 A157241 A157242 * A157244 A157245 A157246

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 11 2022

Status

approved