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

0,5

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

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

Example

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

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 *)

Prog

(PARI) T(n, k)=binomial(-n, k) \\ Charles R Greathouse IV, Feb 06 2017
(Magma) /* As triangle */ [[Binomial(-n, k): k in [0..n]]: n in [0..11]]; // G. C. Greubel, Nov 21 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

Crossrefs

For the unsigned triangle see A059481.

Sequence in context: A213744 A213745 A213808 * A059481 A113592 A271702

Adjacent sequences: A027552 A027553 A027554 * A027556 A027557 A027558

Keyword

sign,tabl,nice,easy

Author

N. J. A. Sloane, Olivier Gérard

Status

approved