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A027555 Triangle of binomial coefficients C(-n,k).

%I A027555 #35 Nov 24 2023 16:12:58
%S A027555 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 A027555 -56,126,-252,462,1,-7,28,-84,210,-462,924,-1716,1,-8,36,-120,330,
%U A027555 -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 A027555 Triangle of binomial coefficients C(-n,k).
%D A027555 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 164.
%D A027555 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2.
%H A027555 T. D. Noe, <a href="/A027555/b027555.txt">Rows n = 0..50 of triangle, flattened</a>
%F A027555 T(n,k) = binomial(-n,k) = (-1)^k*binomial(n+k-1,k). - _R. J. Mathar_, Feb 06 2015
%F A027555 T(n, k) = (-1)^k * RisingFactorial(n, k) / k!. - _Peter Luschny_, Nov 24 2023
%e A027555 Triangle starts:
%e A027555   1;
%e A027555   1, -1;
%e A027555   1, -2,  3;
%e A027555   1, -3,  6, -10;
%e A027555   1, -4, 10, -20, 35;
%e A027555   1, -5, 15, -35, 70, -126;
%e A027555   ...
%p A027555 A027555 := proc(n,k)
%p A027555     (-1)^k*binomial(n+k-1,k) ;
%p A027555 end proc:
%p A027555 seq(seq(A027555(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Feb 06 2015
%t A027555 Flatten[Table[Binomial[-n,k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Apr 30 2012 *)
%o A027555 (PARI) T(n,k)=binomial(-n,k) \\ _Charles R Greathouse IV_, Feb 06 2017
%o A027555 (Magma) /* As triangle */ [[Binomial(-n, k): k in [0..n]]: n in [0..11]]; // _G. C. Greubel_, Nov 21 2017
%o A027555 (SageMath)
%o A027555 def T(n,k):
%o A027555     return (-1)^k * rising_factorial(n, k) // factorial(k)
%o A027555 for n in range(9):
%o A027555     print([T(n, k) for k in range(n+1)])  # _Peter Luschny_, Nov 24 2023
%Y A027555 For the unsigned triangle see A059481.
%K A027555 sign,tabl,nice,easy
%O A027555 0,5
%A A027555 _N. J. A. Sloane_, _Olivier Gérard_