A176013 Triangle, read by rows, T(n, k) = (-1)^n * n!/(k*k!) * binomial(n-1, k-1) * binomial(n, k-1).

-1, 2, 1, -6, -9, -1, 24, 72, 24, 1, -120, -600, -400, -50, -1, 720, 5400, 6000, 1500, 90, 1, -5040, -52920, -88200, -36750, -4410, -147, -1, 40320, 564480, 1317120, 823200, 164640, 10976, 224, 1, -362880, -6531840, -20321280, -17781120, -5334336, -592704, -24192, -324, -1

Offset: 1

Author

Roger L. Bagula, Apr 06 2010

Keywords

• sign
• tabl
• easy
• less

Comments

Row sums are: -1, 3, -16, 121, -1171, 13711, -187468, 2920961, -50948677, 981458011, ...

Examples

			Triangle begins as:
       -1;
        2,        1;
       -6,       -9,        -1;
       24,       72,        24,         1;
     -120,     -600,      -400,       -50,       -1;
      720,     5400,      6000,      1500,       90,       1;
    -5040,   -52920,    -88200,    -36750,    -4410,    -147,     -1;
    40320,   564480,   1317120,    823200,   164640,   10976,    224,    1;
  -362880, -6531840, -20321280, -17781120, -5334336, -592704, -24192, -324, -1;
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A008297.

Programs

Magma

[(-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 08 2021

Mathematica

T[n_, k_] = (-1)^n*n!/(k*k!)*Binomial[n-1, k-1]*Binomial[n, k-1];
Table[T[n, k], {n,12}, {k,n}]//Flatten

Sage

flatten([[(-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 08 2021

Formula

T(n, k) = (-1)^n * n!/(k*k!) * binomial(n-1, k-1) * binomial(n, k-1).

T(n, k) = binomial(n+1, k) * A008297(n, k)/(n+1). - G. C. Greubel, Feb 08 2021

Extensions

Edited by G. C. Greubel, Feb 08 2021