It is known that the following five counting problems lead to the same
integer sequence~ftβ(n): the number of nonequivalent compact Huffman codes of
length~n over an alphabet of t letters, the number of `nonequivalent'
canonical rooted t-ary trees (level-greedy trees) with n~leaves, the number
of `proper' words, the number of bounded degree sequences, and the number of
ways of writing 1=tx1β1β+β―+txnβ1β with integers
0β€x1ββ€x2ββ€β―β€xnβ. In this work, we show that one can
compute this sequence for \textbf{all} n<N with essentially one power series
division. In total we need at most N1+Ξ΅ additions and
multiplications of integers of cN bits, c<1, or N2+Ξ΅ bit
operations, respectively. This improves an earlier bound by Even and Lempel who
needed O(N3) operations in the integer ring or O(N4) bit operations,
respectively