cp's OEIS Frontend

A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

%I A320268 #12 Jan 22 2023 18:17:36
%S A320268 1,0,1,1,2,3,6,9,16,26,44,70,119,189,314,506,830,1336,2186,3522,5737,
%T A320268 9266,15047,24313,39444,63759,103322,167098,270616,437714,708676,
%U A320268 1146390,1855582,3002017,4858429,7860454,12720310,20580764,33303260,53884144,87190964
%N A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.
%C A320268 This is a weaker condition than achirality (cf. A167865).
%C A320268 A rooted tree is series-reduced if every non-leaf node has at least two branches.
%H A320268 Andrew Howroyd, <a href="/A320268/b320268.txt">Table of n, a(n) for n = 1..500</a>
%F A320268 a(1) = 1; a(2) = 0; a(n > 2) = 1 + Sum_{k = 2..n-2} floor((n-1)/k) * a(k).
%e A320268 The a(3) = 1 through a(8) = 9 rooted trees:
%e A320268   (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
%e A320268                (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
%e A320268                         (oo(oo))  (oo(ooo))   (oo(oooo))
%e A320268                                   (ooo(oo))   (ooo(ooo))
%e A320268                                   ((oo)(oo))  (oooo(oo))
%e A320268                                   (o(o(oo)))  (o(o(ooo)))
%e A320268                                               (o(oo)(oo))
%e A320268                                               (o(oo(oo)))
%e A320268                                               (oo(o(oo)))
%t A320268 saum[n_]:=Sum[If[DeleteCases[ptn,1]=={},1,saum[DeleteCases[ptn,1][[1]]]],{ptn,Select[IntegerPartitions[n-1],And[Length[#]!=1,SameQ@@DeleteCases[#,1]]&]}];
%t A320268 Array[saum,20]
%o A320268 (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=3, n, v[n] = 1 + sum(k=2, n-2, (n-1)\k*v[k])); v} \\ _Andrew Howroyd_, Oct 26 2018
%Y A320268 Cf. A001678, A002541, A003238, A010766, A070776, A014668, A126656, A167865, A317099, A317100, A317712, A320222, A320226, A320269.
%K A320268 nonn
%O A320268 1,5
%A A320268 _Gus Wiseman_, Oct 08 2018