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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.

%I A320222 #15 Jan 22 2023 18:54:08
%S A320222 1,1,2,4,9,18,39,78,161,324,658,1316,2657,5314,10668,21347,42777,
%T A320222 85554,171290,342580,685498,1371037,2742733,5485466,10972351,21944711,
%U A320222 43892080,87784323,175574004,351148008,702307038,1404614076,2809249582,5618499824,11237042426
%N A320222 Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.
%C A320222 This is a weaker condition than achirality (cf. A003238).
%H A320222 Andrew Howroyd, <a href="/A320222/b320222.txt">Table of n, a(n) for n = 1..500</a>
%F A320222 a(n) = 1 + Sum_{k = 2..n-1} floor((n-1)/k) * a(k).
%F A320222 a(n) ~ c * 2^n, where c = 0.3270422384018894564479397100499014525700668391191792769625407295138546463... - _Vaclav Kotesovec_, Sep 07 2019
%e A320222 The a(1) = 1 through a(6) = 18 rooted trees:
%e A320222   o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
%e A320222           ((o))  ((oo))   ((ooo))    ((oooo))
%e A320222                  (o(o))   (o(oo))    (o(ooo))
%e A320222                  (((o)))  (oo(o))    (oo(oo))
%e A320222                           (((oo)))   (ooo(o))
%e A320222                           ((o)(o))   (((ooo)))
%e A320222                           ((o(o)))   ((o(oo)))
%e A320222                           (o((o)))   ((oo(o)))
%e A320222                           ((((o))))  (o((oo)))
%e A320222                                      (o(o)(o))
%e A320222                                      (o(o(o)))
%e A320222                                      (oo((o)))
%e A320222                                      ((((oo))))
%e A320222                                      (((o)(o)))
%e A320222                                      (((o(o))))
%e A320222                                      ((o((o))))
%e A320222                                      (o(((o))))
%e A320222                                      (((((o)))))
%t A320222 saue[n_]:=Sum[If[SameQ@@DeleteCases[ptn,1],If[DeleteCases[ptn,1]=={},1,saue[DeleteCases[ptn,1][[1]]]],0],{ptn,IntegerPartitions[n-1]}];
%t A320222 Table[saue[n],{n,15}]
%o A320222 (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 A320222 Cf. A002541, A003238, A010766, A126656, A014668, A167865, A214577, A316782, A317099, A317100, A317712, A320230.
%K A320222 nonn
%O A320222 1,3
%A A320222 _Gus Wiseman_, Oct 07 2018