%I #35 Nov 24 2023 16:12:58

%S 1,1,-1,1,-2,3,1,-3,6,-10,1,-4,10,-20,35,1,-5,15,-35,70,-126,1,-6,21,

%T -56,126,-252,462,1,-7,28,-84,210,-462,924,-1716,1,-8,36,-120,330,

%U -792,1716,-3432,6435,1,-9,45,-165,495,-1287,3003,-6435,12870,-24310,1,-10,55,-220,715,-2002,5005,-11440,24310,-48620,92378

%N Triangle of binomial coefficients C(-n,k).

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 164.

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2.

%H T. D. Noe, <a href="/A027555/b027555.txt">Rows n = 0..50 of triangle, flattened</a>

%F T(n,k) = binomial(-n,k) = (-1)^k*binomial(n+k-1,k). - _R. J. Mathar_, Feb 06 2015

%F T(n, k) = (-1)^k * RisingFactorial(n, k) / k!. - _Peter Luschny_, Nov 24 2023

%e Triangle starts:

%e 1;

%e 1, -1;

%e 1, -2, 3;

%e 1, -3, 6, -10;

%e 1, -4, 10, -20, 35;

%e 1, -5, 15, -35, 70, -126;

%e ...

%p A027555 := proc(n,k)

%p (-1)^k*binomial(n+k-1,k) ;

%p end proc:

%p seq(seq(A027555(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Feb 06 2015

%t Flatten[Table[Binomial[-n,k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Apr 30 2012 *)

%o (PARI) T(n,k)=binomial(-n,k) \\ _Charles R Greathouse IV_, Feb 06 2017

%o (Magma) /* As triangle */ [[Binomial(-n, k): k in [0..n]]: n in [0..11]]; // _G. C. Greubel_, Nov 21 2017

%o (SageMath)

%o def T(n,k):

%o return (-1)^k * rising_factorial(n, k) // factorial(k)

%o for n in range(9):

%o print([T(n, k) for k in range(n+1)]) # _Peter Luschny_, Nov 24 2023

%Y For the unsigned triangle see A059481.

%K sign,tabl,nice,easy

%O 0,5

%A _N. J. A. Sloane_, _Olivier Gérard_