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A053200
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Binomial coefficients C(n,k) reduced modulo n, read by rows ; T(0,0)=0 by convention .
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10
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,13
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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 polynomial-time algorithm for primality testing. - N. J. A. Sloane (njas(AT)research.att.com), Feb 20, 2004
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REFERENCES
| M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Ann. of Math. (2) 160 (2004), no. 2, 781-793.
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LINKS
| T. D. Noe, Rows n=0..100 of triangle, flattened
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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 ;
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MAPLE
| f := n -> seriestolist( series( expand( (1+x)^n ) mod n, x, n+1)); (N. J. A. Sloane)
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CROSSREFS
| Row sums give A053204. Cf. A053201, A053202, A053203, A007318 (Pascal's triangle)
Cf. also A092241.
Cf. A007318.
Sequence in context: A158924 A025426 A204246 * A050870 A103306 A163510
Adjacent sequences: A053197 A053198 A053199 * A053201 A053202 A053203
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KEYWORD
| nonn,tabl,nice
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AUTHOR
| Asher Auel (asher.auel(AT)reed.edu) Dec 12, 1999
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EXTENSIONS
| Corrected by T. D. Noe, Feb 08 2008
Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 29 2008 at the suggestion of R. J. Mathar
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