A176861 Triangle T(n, k) = (-1)^n*(k+1)!*(n-k+1)!*binomial(n+2, k+2)*binomial(n+2, n-k+2) read by rows.

1, -6, -6, 36, 64, 36, -240, -600, -600, -240, 1800, 5760, 8100, 5760, 1800, -15120, -58800, -105840, -105840, -58800, -15120, 141120, 645120, 1411200, 1806336, 1411200, 645120, 141120, -1451520, -7620480, -19595520, -30481920, -30481920, -19595520, -7620480, -1451520

Offset: 0

Author

Roger L. Bagula, Apr 27 2010

Keywords

• sign
• tabl
• easy
• less

Comments

Row sums are: 1, -12, 136, -1680, 23220, -359520, 6201216, -118298880, ...

Examples

			Triangle begins as:
       1;
      -6,     -6;
      36,     64,      36;
    -240,   -600,    -600,    -240;
    1800,   5760,    8100,    5760,    1800;
  -15120, -58800, -105840, -105840,  -58800, -15120;
  141120, 645120, 1411200, 1806336, 1411200, 645120, 141120;
		

References

• F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 576 and 270.

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A132159.

Programs

Magma

[(-1)^n*Factorial(k+1)*Factorial(n-k+1)*Binomial(n+2, k+2)*Binomial(n+2, n-k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2021

Mathematica

T[n_, k_]:= (-1)^n*(k+1)!*(n-k+1)!*Binomial[n+2, k+2]*Binomial[n+2, n-k+2];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Sage

flatten([[(-1)^n*factorial(k+1)*factorial(n-k+1)*binomial(n+2, k+2)*binomial(n+2, n-k+2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2021

Formula

T(n, k) = (-1)^n*(k+1)!*(n-k+1)!*binomial(n+2, k+2)*binomial(n+2, n-k+2).

T(n, k) = (-1)^n * A132159(n+2, k+2) * A132159(n+2, n-k+2). - G. C. Greubel, Feb 07 2021

Extensions

Edited by G. C. Greubel, Feb 07 2021