This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383894 #15 May 28 2025 01:08:42
%S A383894 1,1,2,4,9,19,44,96,220,489,1115,2483,5646,12571,28343,63152,141621,
%T A383894 314330,701327,1552149,3445128,7599990,16789039,36908077
%N A383894 Number of arborescent partitions with exactly n parts.
%C A383894 Equivalently, multisets of subtree sizes of rooted trees with n nodes.
%C A383894 The multiset of subtree sizes of a rooted tree T is the multiset containing the number of nodes of the subtrees rooted at each node of T. Integer partitions obtained this way are called arborescent partitions.
%C A383894 All arborescent partitions are spiny partitions (cf. A383895).
%H A383894 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a383/A383894.java">Java program</a> (github)
%e A383894 The following rooted tree has its multiset of subtree sizes equal to {8, 7, 3, 2, 1, 1, 1, 1}:
%e A383894 o
%e A383894 |
%e A383894 o
%e A383894 /|\
%e A383894 / | \
%e A383894 o o o
%e A383894 / \ |
%e A383894 o o o
%e A383894 The 9 arborescent partitions corresponding to a(5) = 9 are:
%e A383894 (51111), (52111), (52211),
%e A383894 (53111), (53211), (54111),
%e A383894 (54211), (54311), (54321).
%e A383894 The following two non-isomorphic trees have the same multiset of subtree sizes, which is {6, 3, 2, 1, 1, 1}:
%e A383894 o o
%e A383894 / \ /|\
%e A383894 o o o o o
%e A383894 / \ \ |
%e A383894 o o o o
%e A383894 |
%e A383894 o
%Y A383894 Cf. A000081 (number of rooted trees), A382440 (subtree sizes of binary trees), A383895 (spiny partitions).
%K A383894 nonn,more
%O A383894 1,3
%A A383894 _Ludovic Schwob_, May 14 2025
%E A383894 a(18)-a(24) from _Sean A. Irvine_, May 25 2025