Where the Möbius transform and matrix inversion agree.

//Program starts, written in Scilab - a Matlab clone
x=20; // size of matrix, don't change

T=zeros(x,x);

//a = fraction A001790/A046161

a=[1/1; 1/2; 3/8; 5/16; 35/128; 63/256;
231/1024; 429/2048; 6435/32768; 12155/65536;
46189/262144; 88179/524288; 676039/4194304;
1300075/8388608; 5014575/33554432;9694845/67108864;
300540195/2147483648; 583401555/4294967296;
2268783825/17179869184; 4418157975/34359738368]

for n=1:x;
T(n,1)=a(n); //set first column equal to fraction
end
T;

for n=2:x;
  for k=2:n;
    s = 0;
    for i=1:k-1;
      s=s+T(n-i,k-1)-T(n-i,k);//divisibility recurrence
    end
    T(n,k)=s;
  end
end
T;

U=inv(T); //Matrix inverse

//inverse Moebius transform of first column in matrix U
V=zeros(x,x);
for n=1:x;
  for k=1:n;
    if modulo(n,k)==0;
      V(n,k)=U(n/k,1);
      end
  end
end
b=sum(V,'c');
// a is the input and b is the output. We notice that a=b and
// therefore it could perhaps be said that the fraction
// A001790/A046161 is invariant under the divisibility recurrence,
// matrix inversion and the inverse Moebius transform.
C=[a,b]
//
// The divisibility recurrence applied to a sequence is the same
// thing as the Dirichlet convolution of another sequence.
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