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

1, 1, -1, 1, -2, 3, 1, -3, 6, -10, 1, -4, 10, -20, 35, 1, -5, 15, -35, 70, -126, 1, -6, 21, -56, 126, -252, 462, 1, -7, 28, -84, 210, -462, 924, -1716, 1, -8, 36, -120, 330, -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

Offset: 0

Author

N. J. A. Sloane, Olivier Gérard

Keywords

• sign
• tabl
• nice
• easy

Examples

			Triangle starts:
  1;
  1, -1;
  1, -2,  3;
  1, -3,  6, -10;
  1, -4, 10, -20, 35;
  1, -5, 15, -35, 70, -126;
  ...
		

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 164.
• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2.

Links

• T. D. Noe, Rows n = 0..50 of triangle, flattened

Crossrefs

For the unsigned triangle see A059481.

Programs

Magma

/* As triangle */ [[Binomial(-n, k): k in [0..n]]: n in [0..11]]; // G. C. Greubel, Nov 21 2017

Maple

A027555 := proc(n,k)
    (-1)^k*binomial(n+k-1,k) ;
end proc:
seq(seq(A027555(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015

Mathematica

Flatten[Table[Binomial[-n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Apr 30 2012 *)

PARI

T(n,k)=binomial(-n,k) \\ Charles R Greathouse IV, Feb 06 2017

SageMath

def T(n,k):
    return (-1)^k * rising_factorial(n, k) // factorial(k)
for n in range(9):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Nov 24 2023

Formula

T(n,k) = binomial(-n,k) = (-1)^k*binomial(n+k-1,k). - R. J. Mathar, Feb 06 2015

T(n, k) = (-1)^k * RisingFactorial(n, k) / k!. - Peter Luschny, Nov 24 2023