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A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

%I A316696 #13 Feb 07 2020 09:05:03
%S A316696 1,2,4,11,27,80,218,654,1923,5924,18310,58176,186341,606814,1993420,
%T A316696 6618160,22134640
%N A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.
%C A316696 A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.
%H A316696 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%e A316696 The a(4) = 11 rooted trees:
%e A316696   4,
%e A316696   (13),
%e A316696   (22),
%e A316696   (1(12)), (2(11)), (112),
%e A316696   (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111).
%t A316696 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A316696 nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],disjointQ],{ptn,Rest[IntegerPartitions[n]]}],{n}];
%t A316696 Table[Length[nms[n]],{n,10}]
%Y A316696 Matula-Goebel numbers of locally disjoint rooted trees are A316495.
%Y A316696 The case where all leaves are 1's is A316697.
%Y A316696 Lone-child-avoiding locally disjoint rooted trees are A331680.
%Y A316696 Cf. A000669, A001678, A141268, A316473, A316652, A331678, A331686, A331687, A331871, A331872, A331874.
%K A316696 nonn,more
%O A316696 1,2
%A A316696 _Gus Wiseman_, Jul 10 2018
%E A316696 a(16)-a(17) from _Robert Price_, Sep 16 2018
%E A316696 Terminology corrected by _Gus Wiseman_, Feb 06 2020