A127173 - OEIS

%I #18 Feb 20 2022 22:52:23

%S 1,-2,1,-2,0,1,1,-2,0,1,-2,0,0,0,1,4,-2,-2,0,0,1,-2,0,0,0,0,0,1,0,1,0,

%T -2,0,0,0,1,1,0,-2,0,0,0,0,0,1,4,-2,0,0,-2,0,0,0,0,1,-2,0,0,0,0,0,0,0,

%U 0,0,1,-2,4,1,-2,0,-2,0,0,0,0,0,1

%N T(n,k) = A007427(n/k) if k divides n, T(n,k) = 0 otherwise.

%H Andrew Howroyd, <a href="/A127173/b127173.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%F Square of A054525 as lower triangular matrix.

%F A007431(n) = Sum_{k=1, n} k*T(n,k).

%F A007428(n) = Sum_{k=1..n} mu(k)*T(n,k).

%e First few rows of the triangle:

%e    1;

%e   -2,  1;

%e   -2,  0,  1;

%e    1, -2,  0,  1;

%e   -2,  0,  0,  0,  1;

%e    4, -2, -2,  0,  0, 1;

%e   -2,  0,  0,  0,  0, 0, 1;

%e    0,  1,  0, -2,  0, 0, 0, 1;

%e    1,  0, -2,  0,  0, 0, 0, 0, 1;

%e    4, -2,  0,  0, -2, 0, 0, 0, 0, 1;

%e   ...

%o (PARI) \\ here b(n) is A007427(n).

%o b(n)={sumdiv(n, d, moebius(d) * moebius(n/d))}

%o T(n,k)={if(n%k==0, b(n/k), 0)} \\ _Andrew Howroyd_, Feb 20 2022

%Y Row sums are A008683.

%Y Column 1 is A007427.

%Y Cf. A054525, A007431, A007428.

%K tabl,sign

%O 1,2

%A _Gary W. Adamson_, Jan 06 2007

%E Missing a(10)-a(14) and a(56) and beyond from _Andrew Howroyd_, Feb 20 2022