A157219 Triangle T(n, k) = binomial(n*f(n,k), f(n,k)), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 28, 4, 1, 1, 5, 45, 45, 5, 1, 1, 6, 66, 816, 66, 6, 1, 1, 7, 91, 1330, 1330, 91, 7, 1, 1, 8, 120, 2024, 35960, 2024, 120, 8, 1, 1, 9, 153, 2925, 58905, 58905, 2925, 153, 9, 1, 1, 10, 190, 4060, 91390, 2118760, 91390, 4060, 190, 10, 1

Offset

0,5

Links

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

Formula

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

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

Example

Triangle begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,  28,    4,     1;
  1,  5,  45,   45,     5,       1;
  1,  6,  66,  816,    66,       6,     1;
  1,  7,  91, 1330,  1330,      91,     7,    1;
  1,  8, 120, 2024, 35960,    2024,   120,    8,   1;
  1,  9, 153, 2925, 58905,   58905,  2925,  153,   9,  1;
  1, 10, 190, 4060, 91390, 2118760, 91390, 4060, 190, 10, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
T[n_, k_]:= Binomial[n*f[n, k], f[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
f:= func< n, k | k le Floor(n/2) select k else n-k >;
[Binomial(n*f(n, k), f(n, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 10 2022
(Sage)
def f(n, k): return k if (k <= (n//2)) else n-k
flatten([[binomial(n*f(n, k), f(n, k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 10 2022

Crossrefs

Sequence in context: A159623 A143199 A137896 * A167040 A054450 A344610

Adjacent sequences: A157216 A157217 A157218 * A157220 A157221 A157222

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved