Let \(T\) be the unit circle and \(a(t)=|1-\tau|^{2p}f(\tau), \tau\in T\), where \(p\in\mathbb{N}\) and \(f\) is a positive continuous function such that \(\sum_{n\in \mathbb{Z}} |n||f_n|<\infty\), and \(f_n\) are the Fourier coefficients. \textit{P. Rambour} and \textit{A. Seghier} [C. R. Acad. Sci., Paris 335, No. 8, 705--710 (2002; Zbl 1012.65025); erratum: ibid. 336, 399--400 (2003); Integral Equations Oper. Theory 50, No. 1, 83--114 (2004; Zbl 1069.47027)] discussed the asymptotic behavior of the \(j,k\) entry \([T_n^{-1} (a)]_{j,k}\) as \(n\to\infty\) of the Toeplitz \((n+1)\times (n+1)\) matrix \(T_n^{-1} (a)=(T_n (a))^{-1},\;T_n (a)=(a_{j-k})_{j,k=0}^n\). They established that \[ [T_n^{-1} (a)]_{[nx],[ny]}= \frac{1}{f(1)}G_p(x,y)n^{2p-1}+o(n^{2p-1}), \] as \(n\to\infty\) uniformly with respect to \(x\) and \(y\) in \([0,1]\). They also remarked that \(G_1(x,y)=x(1-y), G_2(x,y)=\frac 16 x^2(1-y)^2 (3y-x-2xy).\) In the present paper, the author gives an elegant method for the calculation of \(G_p(x,y)\) for any \(p\in \mathbb{N}\). Theorem. For \(0\leq x\leq 1\) and \(y\geq \max (x,1-x)\), \[ G_p(x,y)=\frac{x^py^p}{[(p-1)!]^2}\int_y^1\frac{(t-x)^{p-1}(t-y)^{p-1}}{t^{2p}}dt. \] The asymptotic behavior of the trace and sum of all entries are also considered.