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A317100
Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.
4
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 01 2018
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018
STATUS
approved