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A027555 Triangle of binomial coefficients C(-n,k). 16
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 (list; table; graph; refs; listen; history; text; internal format)
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

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

CROSSREFS

For the unsigned triangle see A059481.

Sequence in context: A213745 A213808 A059481 * A113592 A271702 A292915

Adjacent sequences:  A027552 A027553 A027554 * A027556 A027557 A027558

KEYWORD

sign,tabl,nice,easy

AUTHOR

N. J. A. Sloane, Olivier Gérard

STATUS

approved

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Last modified November 15 11:35 EST 2018. Contains 317238 sequences. (Running on oeis4.)