  We present an improved version of a previous efficient algorithm that
computes the number $D(n)$ of zero-free graphical degree sequences of length
$n$. A main ingredient of the improvement lies in a more efficient way to
compute the function $P(N,k,l,s)$ of Barnes and Savage. We further show that
the algorithm can be easily adapted to compute the $D(i)$ values for all $i\le
n$ in a single run. Theoretical analysis shows that the new algorithm to
compute all $D(i)$ values for $i\le n$ is a constant times faster than the
previous algorithm to compute a single $D(n)$. Experimental evaluations show
that the constant of improvement is about 10. We also perform simulations to
estimate the asymptotic order of $D(n)$ by generating uniform random samples
from the set of $E(n)$ integer partitions of fixed length $n$ with even sum and
largest part less than $n$ and computing the proportion of them that are
graphical degree sequences. The known numerical results of $D(n)$ for $n\le
290$ together with the known bounds of $D(n)$ and simulation results allow us
to make an informed guess about its unknown asymptotic order. The techniques
for the improved algorithm can be applied to compute other similar functions
that count the number of graphical degree sequences of various classes of
graphs of order $n$ and that all involve the function $P(N,k,l,s)$.
