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A320222
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Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.
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12
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1, 1, 2, 4, 9, 18, 39, 78, 161, 324, 658, 1316, 2657, 5314, 10668, 21347, 42777, 85554, 171290, 342580, 685498, 1371037, 2742733, 5485466, 10972351, 21944711, 43892080, 87784323, 175574004, 351148008, 702307038, 1404614076, 2809249582, 5618499824, 11237042426
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OFFSET
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1,3
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COMMENTS
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This is a weaker condition than achirality (cf. A003238).
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LINKS
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FORMULA
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a(n) = 1 + Sum_{k = 2..n-1} floor((n-1)/k) * a(k).
a(n) ~ c * 2^n, where c = 0.3270422384018894564479397100499014525700668391191792769625407295138546463... - Vaclav Kotesovec, Sep 07 2019
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EXAMPLE
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The a(1) = 1 through a(6) = 18 rooted trees:
o (o) (oo) (ooo) (oooo) (ooooo)
((o)) ((oo)) ((ooo)) ((oooo))
(o(o)) (o(oo)) (o(ooo))
(((o))) (oo(o)) (oo(oo))
(((oo))) (ooo(o))
((o)(o)) (((ooo)))
((o(o))) ((o(oo)))
(o((o))) ((oo(o)))
((((o)))) (o((oo)))
(o(o)(o))
(o(o(o)))
(oo((o)))
((((oo))))
(((o)(o)))
(((o(o))))
((o((o))))
(o(((o))))
(((((o)))))
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MATHEMATICA
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saue[n_]:=Sum[If[SameQ@@DeleteCases[ptn, 1], If[DeleteCases[ptn, 1]=={}, 1, saue[DeleteCases[ptn, 1][[1]]]], 0], {ptn, IntegerPartitions[n-1]}];
Table[saue[n], {n, 15}]
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PROG
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(PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=2, n-1, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018
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CROSSREFS
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Cf. A002541, A003238, A010766, A126656, A014668, A167865, A214577, A316782, A317099, A317100, A317712, A320230.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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