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

1, 3, 5, 12, 17, 41, 65, 144, 262, 533, 1025, 2110, 4097, 8261, 16407, 32928, 65537, 131384, 262145, 524854, 1048647, 2098181, 4194305, 8390924, 16777234, 33558533, 67109132, 134226070, 268435457, 536887919, 1073741825, 2147516736, 4294968327, 8590000133

Offset: 1

Author

Gus Wiseman, Aug 01 2018

Keywords

• nonn

Comments

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.

Examples

			The a(4) = 12 trees:
  (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
  (1222),
  (1122), ((12)(12)),
  (1112),
  (1233),
  (1223),
  (1123),
  (1234).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A001678, A003238, A052409, A052410, A067824, A167865, A168532, A214577, A289078, A294336, A316782, A317099.

Programs

Mathematica

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
b[n_]:=1+Sum[b[n/d],{d,Rest[Divisors[n]]}];
a[n_]:=Sum[b[GCD@@Length/@Split[ptn]],{ptn,allnorm[n]}];
Array[a,10]

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

Formula

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 07 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018