1,2

Unsigned elements consist of only 0's, 1's, and 2's.

T(n,1) = +2 when n = 2*p where p is an odd prime.

T(n,1) = -2 when n is an even square-free number with an odd number of prime divisors.

A123709(n) = number of nonzero terms in row n = 2^(m+1) - 1 when n is an odd number with exactly m distinct prime factors.

Sum_{k=1..n} T(n,k) = moebius(n).

Sum_{k=1..n} T(n,k)*k = 0 for n>1.

Sum_{k=1..n} T(n,k)*k^2 = 2*phi(n) for n>1 where phi(n)=A000010(n).

Sum_{k=1..n} T(n,k)*k^3 = 6*A102309(n) for n>1 where A102309(n)=Sum[d|n, moebius(d)*C(n/d,2) ].

Sum_{k=1..n} T(n,k)*k*2^(k-1) = A085411(n) = Sum_{d|n} mu(n/d)*(d+1)*2^(d-2) = total number of parts in all compositions of n into relatively prime parts.

Triangle begins:

1;

-2, 1;

-1,-1, 1;

1,-1,-1, 1;

-1, 0, 0,-1, 1;

2, 0,-1, 0,-1, 1;

-1, 0, 0, 0, 0,-1, 1;

0, 0, 1,-1, 0, 0,-1, 1;

0, 1,-1, 0, 0, 0, 0,-1, 1;

2,-1, 0, 1,-1, 0, 0, 0,-1, 1;

-1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1;

-1, 1, 1,-1, 1,-1, 0, 0, 0, 0,-1, 1;

-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1;

2,-1, 0, 0, 0, 1,-1, 0, 0, 0, 0, 0,-1, 1;

1, 1,-1, 1,-1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1; ...

(PARI) T(n, k)=(matrix(n, n, r, c, if(r>=c, floor(r/c)))^-1)[n, k]

Cf. A102309, A085411; A123707, A123708, A123709.

Sequence in context: A056980 A005094 A121372 this_sequence A025452 A054977 A078315

Adjacent sequences: A123703 A123704 A123705 this_sequence A123707 A123708 A123709

sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 09 2006