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

1,2

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.

Links

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

Formula

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

Example

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).

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]

Prog

(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

Crossrefs

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

Sequence in context: A299490 A361274 A126471 * A199932 A305552 A226652

Adjacent sequences: A317097 A317098 A317099 * A317101 A317102 A317103

Keyword

nonn

Author

Gus Wiseman, Aug 01 2018

Extensions

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

Status

approved