A169656 Triangle, read by rows, T(n, k) = (-1)^n*(n!/k!)^2*binomial(n-1, k-1).

-1, 4, 1, -36, -18, -1, 576, 432, 48, 1, -14400, -14400, -2400, -100, -1, 518400, 648000, 144000, 9000, 180, 1, -25401600, -38102400, -10584000, -882000, -26460, -294, -1, 1625702400, 2844979200, 948326400, 98784000, 3951360, 65856, 448, 1

Offset: 1

Author

Roger L. Bagula, Apr 05 2010

Keywords

• sign
• tabl

Comments

Row sums are: {-1, 5, -55, 1057, -31301, 1319581, -74996755, 5521809665, -510921831817, 58003632177301, ...}.

Examples

			Triangle begins as:
         -1;
          4,         1;
        -36,       -18,        -1;
        576,       432,        48,       1;
     -14400,    -14400,     -2400,    -100,     -1;
     518400,    648000,    144000,    9000,    180,    1;
  -25401600, -38102400, -10584000, -882000, -26460, -294, -1;
		

Links

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

Crossrefs

Cf. A008297.

Programs

GAP

F:=Factorial;; Flat(List([1..10], n-> List([1..n], k-> (-1)^n*(F(n)/F(k) )^2*Binomial(n-1, k-1) ))); # G. C. Greubel, Nov 28 2019

Magma

F:=Factorial; [(-1)^n*(F(n)/F(k))^2*Binomial(n-1, k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 28 2019

Maple

seq(seq( (-1)^n*(n!/k!)^2*binomial(n-1, k-1), k=1..n), n=1..10); # G. C. Greubel, Nov 28 2019

Mathematica

T[n_, k_]:= (-1)^n*(n!/k!)^2*Binomial[n-1, k-1]; Table[T[n, k], {n,10}, {k,n}]//Flatten

PARI

T(n,k) = (-1)^n*(n!/k!)^2*binomial(n-1, k-1); \\ G. C. Greubel, Nov 28 2019

Sage

f=factorial; [[(-1)^n*(f(n)/f(k))^2*binomial(n-1, k-1) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 28 2019

Formula

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

Extensions

Edited by G. C. Greubel, Nov 28 2019