Calculate the determinant of the matrices $a_{ij}=\frac{1}{i+j-1}$ and $b_{ij}=\frac{1}{i+j}$? I would like to know if there is any formula for calculating determinants of the following symmetric matrices:
$$
A=[a_{ij}]_{n\times n},\qquad a_{ij}=\frac{1}{i+j-1}, 
$$
and
$$
B=[b_{ij}]_{n\times n},\qquad b_{ij}=\frac{1}{i+j},\qquad 
$$
where $1\leq i,j\leq n$.
If there are no formula for the determinants, is it true that the determinant of these matrices are never zero.
Thanks! 

For your $a_{ij}$ matrix, the answer is given by the reciprocal of this sequence in the OEIS (so in the $n=1$ case the determinant is 1, in the $n=2$ case it's 1/12, in the $n=3$ case it's 1/2160, and so on.)  For your $b_{ij}$ matrix, the answer is given by the reciprocal of this sequence.)  According to the OEIS the formulas you want are:
$$\det A_n =   {\prod_{k=0}^{n-1} k!^2 \over n^n \prod_{k=1}^{n-1} (n^2-k^2)^{n-k}}$$
and $\det B_n = \det A_n / {2n \choose n}$.
A note in the OEIS says that these formulas are proven from results on Cauchy matrices.
