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A317580
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Number of unlabeled rooted identity trees with n nodes and a distinguished leaf.
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3
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1, 1, 1, 3, 5, 12, 28, 66, 153, 367, 880, 2121, 5127, 12441, 30248, 73746, 180077, 440571, 1079438, 2648511, 6506170, 16001256, 39393173, 97074140, 239419963, 590972968, 1459808862, 3608483107, 8925476591, 22090139751, 54702648393, 135533335933, 335967782916
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OFFSET
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1,4
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COMMENTS
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Total number of leaves in all rooted identity trees with n nodes. - Andrew Howroyd, Aug 28 2018
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LINKS
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FORMULA
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EXAMPLE
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The a(6) = 12 rooted identity trees with a distinguished leaf:
(((((O))))),
(((O(o)))), (((o(O)))),
((O((o)))), ((o((O)))),
(O(((o)))), (o(((O)))),
((O)((o))), ((o)((O))),
(O(o(o))), (o(O(o))), (o(o(O))).
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MATHEMATICA
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urit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[urit/@ptn]], UnsameQ@@#&], {ptn, IntegerPartitions[n-1]}];
Table[Sum[Length[Flatten[{t/.{}->1}]], {t, urit[n]}], {n, 10}]
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PROG
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(PARI) WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)}
seq(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p -> subst(deriv(p), y, 1), v)} \\ Andrew Howroyd, Aug 28 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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