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A318691
Number of series-reduced powerful uniform rooted trees with n nodes.
4
1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 6, 1, 5, 4, 8, 1, 11, 1, 15, 6, 13, 1, 26, 3, 24, 9, 36, 1, 50, 1, 58, 14, 67, 7, 107, 1, 105, 25, 160, 1, 213, 1, 245, 45, 291, 1, 443, 5, 492, 68, 644, 1, 851, 15, 1019, 106, 1263, 1, 1785, 1, 1986, 189, 2592, 26, 3426, 1, 4071, 292
OFFSET
1,7
COMMENTS
A series-reduced powerful uniform rooted tree with n nodes is a powerful uniform multiset (all multiplicities are equal to the same number > 1) of series-reduced powerful uniform rooted trees with a total of n-1 nodes.
LINKS
FORMULA
a(p+1) = 1 for prime p. - Andrew Howroyd, Dec 09 2020
EXAMPLE
The a(19) = 11 series-reduced powerful uniform rooted trees with 19 nodes:
(((ooo)(ooo))((ooo)(ooo)))
((oo(oo)(oo))(oo(oo)(oo)))
((oo)(oo)(oo)(oo)(oo)(oo))
((oo)(oo)(ooooo)(ooooo))
((ooo)(ooo)(oooo)(oooo))
(oo(oo)(oo)(oooo)(oooo))
((ooooo)(ooooo)(ooooo))
(ooo(oooo)(oooo)(oooo))
((oooooooo)(oooooooo))
(oo(ooooooo)(ooooooo))
(oooooooooooooooooo)
MATHEMATICA
rurt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]], And[Min@@Length/@Split[#]>=2, SameQ@@Length/@Split[#]]&], {ptn, IntegerPartitions[n-1]}]];
Table[Length[rurt[n]], {n, 10}]
PROG
(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=vector(n)); v[1]=1; for(n=1, n-1, my(u=WeighT(v[1..n])); v[n+1] = sumdiv(n, d, u[d]) - u[n]); v} \\ Andrew Howroyd, Dec 09 2020
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 31 2018
EXTENSIONS
Terms a(51) and beyond from Andrew Howroyd, Dec 09 2020
STATUS
approved