A053200 Binomial coefficients C(n,k) reduced modulo n, read by rows; T(0,0)=0 by convention.

0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 3, 2, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 0, 0, 3, 0, 0, 3, 0, 0, 1, 1, 0, 5, 0, 0, 2, 0, 0, 5, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 4, 3, 0, 0, 0, 3, 4, 6, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

0,13

Comments

Pascal's triangle read by rows, where row n is read mod n.

A number n is a prime if and only if (1+x)^n == 1+x^n (mod n), i.e., if and only if the n-th row is 1,0,0,...,0,1. This result underlies the proof of Agrawal, Kayal and Saxena that there is a polynomial-time algorithm for primality testing. - N. J. A. Sloane, Feb 20 2004

A020475(n) = number of zeros in n-th row, for n > 0. - Reinhard Zumkeller, Jan 01 2013

Links

T. D. Noe, Rows n = 0..100 of triangle, flattened

M. Agrawal, N. Kayal & N. Saxena, PRIMES is in P, Annals of Maths., 160:2 (2004), pp. 781-793. [alternate link]

Index entries for triangles and arrays related to Pascal's triangle

Example

Row 4 = 1 mod 4, 4 mod 4, 6 mod 4, 4 mod 4, 1 mod 4 = 1, 0, 2, 0, 1.
Triangle begins:
  0;
  0,0;
  1,0,1;
  1,0,0,1;
  1,0,2,0,1;
  1,0,0,0,0,1;
  1,0,3,2,3,0,1;
  1,0,0,0,0,0,0,1;
  1,0,4,0,6,0,4,0,1;
  1,0,0,3,0,0,3,0,0,1;
  1,0,5,0,0,2,0,0,5,0,1;
  1,0,0,0,0,0,0,0,0,0,0,1;
  1,0,6,4,3,0,0,0,3,4,6,0,1;
  1,0,0,0,0,0,0,0,0,0,0,0,0,1;

Maple

f := n -> seriestolist( series( expand( (1+x)^n ) mod n, x, n+1)); # N. J. A. Sloane

Mathematica

Flatten[Join[{0}, Table[Mod[Binomial[n, Range[0, n]], n], {n, 20}]]] (* Harvey P. Dale, Apr 29 2013 *)

Prog

(Haskell)
a053200 n k = a053200_tabl !! n !! k
a053200_row n = a053200_tabl !! n
a053200_tabl = [0] : zipWith (map . flip mod) [1..] (tail a007318_tabl)
-- Reinhard Zumkeller, Jul 10 2015, Jan 01 2013
(PARI) T(n, k)=if(n, binomial(n, k)%n, 0) \\ Charles R Greathouse IV, Feb 07 2017

Crossrefs

Row sums give A053204. Cf. A053201, A053202, A053203, A007318 (Pascal's triangle).

Cf. also A092241.

Cf. A053214 (central terms, apart from initial 1).

Sequence in context: A025426 A269244 A204246 * A050870 A103306 A269249

Adjacent sequences: A053197 A053198 A053199 * A053201 A053202 A053203

Keyword

nonn,tabl,nice

Author

Asher Auel, Dec 12 1999

Extensions

Corrected by T. D. Noe, Feb 08 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

Status

approved