A143239 Triangle read by rows, A126988 * A128407 as infinite lower triangular matrices.

1, 2, -1, 3, 0, -1, 4, -2, 0, 0, 5, 0, 0, 0, -1, 6, -3, -2, 0, 0, 1, 7, 0, 0, 0, 0, 0, -1, 8, -4, 0, 0, 0, 0, 0, 0, 9, 0, -3, 0, 0, 0, 0, 0, 0, 10, -5, 0, 0, -2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 14, -7, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1

Offset

1,2

Comments

Row sums = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4,...); as a consequence of the Dedekind-Liouville rule illustrated in the example and on p. 137 of "Concrete Mathematics".

References

Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 2nd ed.; Addison-Wesley, 1994, p. 137.

Links

Table of n, a(n) for n=1..105.

Formula

Triangle read by rows generated from the Dedekind-Liouville rule: T(n,k) = mu(k)*(n/k) if k divides n. T(n,k) = 0 if k is not a divisor of n. Equals A126988 * A128407

Example

First few rows of the triangle are:
1;
2, -1;
3, 0, -1;
4, -2, 0, 0;
5, 0, 0, 0, -1;
6, -3, -2, 0, 0, 1;
7, 0, 0, 0, 0, 0, -1;
8, -4, 0, 0, 0, 0, 0, 0;
9, 0, -3, 0, 0, 0, 0, 0, 0;
10, -5, 0, 0, -2, 0, 0, 0, 0, 1;
11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0;
13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
14, -7, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1;
...
Row 12 = (12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0) since (Cf. A126988 - the divisors of 12 are (12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1) and applying mu(k) * (nonzero terms), we get (1*12, (-1)*6, (-1)*4, 1*2) sum = 4 = phi(12).

Crossrefs

Cf. A000010, A128407, A126988, A008683.

Sequence in context: A221642 A158906 A309229 * A158951 A126988 A280499

Adjacent sequences: A143236 A143237 A143238 * A143240 A143241 A143242

Keyword

tabl,sign

Author

Gary W. Adamson, Aug 01 2008

Status

approved