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A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.

%I A317100 #13 Sep 07 2019 04:15:22
%S A317100 1,3,5,12,17,41,65,144,262,533,1025,2110,4097,8261,16407,32928,65537,
%T A317100 131384,262145,524854,1048647,2098181,4194305,8390924,16777234,
%U A317100 33558533,67109132,134226070,268435457,536887919,1073741825,2147516736,4294968327,8590000133
%N A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.
%C A317100 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.
%H A317100 Andrew Howroyd, <a href="/A317100/b317100.txt">Table of n, a(n) for n = 1..200</a>
%F A317100 a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, Sep 07 2019
%e A317100 The a(4) = 12 trees:
%e A317100   (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
%e A317100   (1222),
%e A317100   (1122), ((12)(12)),
%e A317100   (1112),
%e A317100   (1233),
%e A317100   (1223),
%e A317100   (1123),
%e A317100   (1234).
%t A317100 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A317100 b[n_]:=1+Sum[b[n/d],{d,Rest[Divisors[n]]}];
%t A317100 a[n_]:=Sum[b[GCD@@Length/@Split[ptn]],{ptn,allnorm[n]}];
%t A317100 Array[a,10]
%o A317100 (PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n]=2^(n-1) + sumdiv(n, d, v[d])); v} \\ _Andrew Howroyd_, Aug 19 2018
%Y A317100 Cf. A001678, A003238, A052409, A052410, A067824, A167865, A168532, A214577, A289078, A294336, A316782, A317099.
%K A317100 nonn
%O A317100 1,2
%A A317100 _Gus Wiseman_, Aug 01 2018
%E A317100 Terms a(21) and beyond from _Andrew Howroyd_, Aug 19 2018