Summary: The Taylor series expansions of the Kurepa function \(K(a + z)\), \(a \geq 0\), and numerical determination of their coefficients \(b_\nu (a)\) for \(a = 0\) and \(a = 1\) are given. An asymptotic behaviour of \(b_\nu (a)\) as well as that \(|b_\nu (a)/b_{\nu + 1} (a) |\sim a + 1\), when \(\nu \to \infty\), are shown. Using this fact, a transformation of series with much faster convergence is done. Numerical values of coefficients in such a transformed series for \(a = 0\) and \(a = 1\) are given with 30 decimal digits. Also, the Chebyshev expansions of \(K(1 + z)\) and \(1/K(1 + z)\) are obtained.