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A320222 Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal. 12

%I

%S 1,1,2,4,9,18,39,78,161,324,658,1316,2657,5314,10668,21347,42777,

%T 85554,171290,342580,685498,1371037,2742733,5485466,10972351,21944711,

%U 43892080,87784323,175574004,351148008,702307038,1404614076,2809249582,5618499824,11237042426

%N Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.

%C This is a weaker condition than achirality (cf. A003238).

%H Andrew Howroyd, <a href="/A320222/b320222.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = 1 + Sum_{k = 2...n-1} floor((n-1)/k) * a(k).

%F a(n) ~ c * 2^n, where c = 0.3270422384018894564479397100499014525700668391191792769625407295138546463... - _Vaclav Kotesovec_, Sep 07 2019

%e The a(1) = 1 through a(6) = 18 rooted trees:

%e o (o) (oo) (ooo) (oooo) (ooooo)

%e ((o)) ((oo)) ((ooo)) ((oooo))

%e (o(o)) (o(oo)) (o(ooo))

%e (((o))) (oo(o)) (oo(oo))

%e (((oo))) (ooo(o))

%e ((o)(o)) (((ooo)))

%e ((o(o))) ((o(oo)))

%e (o((o))) ((oo(o)))

%e ((((o)))) (o((oo)))

%e (o(o)(o))

%e (o(o(o)))

%e (oo((o)))

%e ((((oo))))

%e (((o)(o)))

%e (((o(o))))

%e ((o((o))))

%e (o(((o))))

%e (((((o)))))

%t saue[n_]:=Sum[If[SameQ@@DeleteCases[ptn,1],If[DeleteCases[ptn,1]=={},1,saue[DeleteCases[ptn,1][[1]]]],0],{ptn,IntegerPartitions[n-1]}];

%t Table[saue[n],{n,15}]

%o (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

%Y Cf. A002541, A003238, A010766, A126656, A014668, A167865, A214577, A316782, A317099, A317100, A317712, A320230.

%K nonn

%O 1,3

%A _Gus Wiseman_, Oct 07 2018

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Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)