A174298 Triangle T(n, k) = binomial(n, k)*( n!/k! if floor(n/2) >= k otherwise n!/(n-k)! ), read by rows.

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 72, 96, 24, 120, 600, 600, 600, 600, 120, 720, 4320, 5400, 2400, 5400, 4320, 720, 5040, 35280, 52920, 29400, 29400, 52920, 35280, 5040, 40320, 322560, 564480, 376320, 117600, 376320, 564480, 322560, 40320, 362880, 3265920, 6531840, 5080320, 1905120, 1905120, 5080320, 6531840, 3265920, 362880

Offset: 0

Author

Roger L. Bagula, Mar 15 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
      1;
      1,      1;
      2,      4,      2;
      6,     18,     18,      6;
     24,     96,     72,     96,     24;
    120,    600,    600,    600,    600,    120;
    720,   4320,   5400,   2400,   5400,   4320,    720;
   5040,  35280,  52920,  29400,  29400,  52920,  35280,   5040;
  40320, 322560, 564480, 376320, 117600, 376320, 564480, 322560, 40320;
		

Links

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

Crossrefs

Cf. A295383.

Programs

Magma

A174298:= func< n,k | Floor(n/2) gt k select Factorial(n-k)*Binomial(n,k)^2 else Factorial(k)*Binomial(n,k)^2 >;
[A174298(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 24 2021

Mathematica

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

Sage

def A174298(n,k): return binomial(n,k)^2*( factorial(n-k) if ((n//2) > k-1) else factorial(k))
flatten([[A174298(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 24 2021

Formula

T(n, k) = binomial(n, k)*( n!/k! if floor(n/2) >= k otherwise n!/(n-k)! ).

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

T(n, k) = binomial(n, k)^2*( (n-k)! if floor(n/2) >= k otherwise k! ).

T(n, 0) = T(n, n) = n!.

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

T(2*n, n) = (-1)^n*A295383(n). (End)

Extensions

Edited by G. C. Greubel, Nov 24 2021