cp's OEIS Frontend

A127173 T(n,k) = A007427(n/k) if k divides n, T(n,k) = 0 otherwise.

%I A127173 #18 Feb 20 2022 22:52:23
%S A127173 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 A127173 -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 A127173 0,0,1,-2,4,1,-2,0,-2,0,0,0,0,0,1
%N A127173 T(n,k) = A007427(n/k) if k divides n, T(n,k) = 0 otherwise.
%H A127173 Andrew Howroyd, <a href="/A127173/b127173.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A127173 Square of A054525 as lower triangular matrix.
%F A127173 A007431(n) = Sum_{k=1, n} k*T(n,k).
%F A127173 A007428(n) = Sum_{k=1..n} mu(k)*T(n,k).
%e A127173 First few rows of the triangle:
%e A127173    1;
%e A127173   -2,  1;
%e A127173   -2,  0,  1;
%e A127173    1, -2,  0,  1;
%e A127173   -2,  0,  0,  0,  1;
%e A127173    4, -2, -2,  0,  0, 1;
%e A127173   -2,  0,  0,  0,  0, 0, 1;
%e A127173    0,  1,  0, -2,  0, 0, 0, 1;
%e A127173    1,  0, -2,  0,  0, 0, 0, 0, 1;
%e A127173    4, -2,  0,  0, -2, 0, 0, 0, 0, 1;
%e A127173   ...
%o A127173 (PARI) \\ here b(n) is A007427(n).
%o A127173 b(n)={sumdiv(n, d, moebius(d) * moebius(n/d))}
%o A127173 T(n,k)={if(n%k==0, b(n/k), 0)} \\ _Andrew Howroyd_, Feb 20 2022
%Y A127173 Row sums are A008683.
%Y A127173 Column 1 is A007427.
%Y A127173 Cf. A054525, A007431, A007428.
%K A127173 tabl,sign
%O A127173 1,2
%A A127173 _Gary W. Adamson_, Jan 06 2007
%E A127173 Missing a(10)-a(14) and a(56) and beyond from _Andrew Howroyd_, Feb 20 2022