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%I #9 Aug 01 2018 20:37:28
%S 1,3,4,9,8,19,16,35,35,54,57,113,102,155,189,279,298,447,491,702,813,
%T 1063,1256,1759,1967,2542,3050,3902,4566,5882,6843,8676,10205,12612,
%U 14908,18608,21638,26510,31292,38150,44584,54185,63262,76308,89371,106818,124755
%N Number of series-reduced planted achiral trees whose leaves span an initial interval of positive integers appearing with multiplicities an integer partition of n.
%C In these trees, achiral means that all branches directly under any given node that is not a leaf or a cover of leaves are equal, and series-reduced means that every node that is not a leaf or a cover of leaves has at least two branches.
%e The a(4) = 9 trees:
%e (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
%e (1112),
%e (1122), ((12)(12)),
%e (1123),
%e (1234).
%e The a(6) = 19 trees:
%e (111111), ((111)(111)), (((1)(1)(1))((1)(1)(1))), ((11)(11)(11)), (((1)(1))((1)(1))((1)(1))), ((1)(1)(1)(1)(1)(1)),
%e (111112),
%e (111122), ((112)(112)),
%e (111123),
%e (111222), ((12)(12)(12)),
%e (111223),
%e (111234),
%e (112233), ((123)(123)),
%e (112234),
%e (112345),
%e (123456).
%t b[n_]:=1+Sum[b[n/d],{d,Rest[Divisors[n]]}];
%t a[n_]:=Sum[b[GCD@@Length/@Split[ptn]],{ptn,IntegerPartitions[n]}];
%t Array[a,30]
%Y Cf. A001678, A003238, A052409, A052410, A067824, A167865, A168532, A214577, A289078, A294336, A316782, A317100.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 01 2018
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