A167040 Triangle T(n, k) = (n-k)^n * binomial(n, n-k) for n < 2*k, k^n * binomial(n, k) for n >= 2*k with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 96, 4, 1, 1, 5, 320, 320, 5, 1, 1, 6, 960, 14580, 960, 6, 1, 1, 7, 2688, 76545, 76545, 2688, 7, 1, 1, 8, 7168, 367416, 4587520, 367416, 7168, 8, 1, 1, 9, 18432, 1653372, 33030144, 33030144, 1653372, 18432, 9, 1

Offset

0,5

Comments

Row sums are: 1, 2, 4, 8, 106, 652, 16514, 158482, 5336706, 69403916, 2915603362, ...

Links

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

Formula

From G. C. Greubel, Feb 24 2021: (Start)

T(n, k) = (n-k)^n * binomial(n, n-k) for n < 2*k, k^n * binomial(n, k) for n >= 2*k with T(n, 0) = T(n, n) = 1.

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

Example

Triangle begins as:
  1;
  1,  1;
  1,  2,     1;
  1,  3,     3,       1;
  1,  4,    96,       4,         1;
  1,  5,   320,     320,         5,          1;
  1,  6,   960,   14580,       960,          6,         1;
  1,  7,  2688,   76545,     76545,       2688,         7,       1;
  1,  8,  7168,  367416,   4587520,     367416,      7168,       8,     1;
  1,  9, 18432, 1653372,  33030144,   33030144,   1653372,   18432,     9,  1;
  1, 10, 46080, 7085880, 220200960, 2460937500, 220200960, 7085880, 46080, 10, 1;

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, If[n>=2*k, k^n*Binomial[n, k], (n-k)^n*Binomial[n, n-k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 24 2021 *)

Prog

(Sage)
@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n > 2*k-1): return k^n*binomial(n, k)
    else: return (n-k)^n*binomial(n, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 24 2021
(Magma)
function T(n, k)
  if k eq 0 or k eq n then return 1;
  elif n lt 2*k then return (n-k)^n*Binomial(n, n-k);
  else return k^n*Binomial(n, k);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 24 2021

Crossrefs

Sequence in context: A143199 A137896 A157219 * A054450 A344610 A337009

Adjacent sequences: A167037 A167038 A167039 * A167041 A167042 A167043

Keyword

nonn,tabl,easy

Author

Roger L. Bagula and Mats Granvik, Oct 27 2009

Extensions

Edited by G. C. Greubel, Feb 24 2021

Status

approved