A317099 - OEIS

%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