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]
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
KEYWORD
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