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
KEYWORD
AUTHOR
STATUS
approved