One matrix equality

In this article a generalization is given of the results existing in the paper [RZhMat, 1968, 1B712]. In the latter the matrix equality
$$ A_n=(-1)^{\frac{n+1}{2}}\biggr[\biggr(\frac{n-3}{2}\biggl)!\biggl]^2A_3^{\frac{n-1}{2}}+(n-1)(n-2)A_{n-2}, $$
is derived, where the elements of matrix $A_k$ are certain linear combinations of the interpolation coefficients of the Lagrange central-difference formula for the second derivative with pattern $K$, and its validity is asserted for $n=5,7,9$, and $11$, which can be established by direct calculation. In the present article it is proved that the matrix equality written above holds for any odd $n$. Matrices of type $A_n$ are encountered when applying the method of lines to certain boundary-value problems in appropriate systems of ordinary differential equations.