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 A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal. 8
 1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 44, 70, 119, 189, 314, 506, 830, 1336, 2186, 3522, 5737, 9266, 15047, 24313, 39444, 63759, 103322, 167098, 270616, 437714, 708676, 1146390, 1855582, 3002017, 4858429, 7860454, 12720310, 20580764, 33303260, 53884144, 87190964 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS This is a weaker condition than achirality (cf. A167865). A rooted tree is series-reduced if every non-leaf node has at least two branches. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..500 FORMULA a(1) = 1; a(2) = 0; a(n > 2) = 1 + Sum_{k = 2...n-2} floor((n-1)/k) * a(k). EXAMPLE The a(3) = 1 through a(8) = 9 rooted trees:   (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)                (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))                         (oo(oo))  (oo(ooo))   (oo(oooo))                                   (ooo(oo))   (ooo(ooo))                                   ((oo)(oo))  (oooo(oo))                                   (o(o(oo)))  (o(o(ooo)))                                               (o(oo)(oo))                                               (o(oo(oo)))                                               (oo(o(oo))) MATHEMATICA saum[n_]:=Sum[If[DeleteCases[ptn, 1]=={}, 1, saum[DeleteCases[ptn, 1][[1]]]], {ptn, Select[IntegerPartitions[n-1], And[Length[#]!=1, SameQ@@DeleteCases[#, 1]]&]}]; Array[saum, 20] PROG (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 CROSSREFS Cf. A001678, A002541, A003238, A010766, A070776, A014668, A126656, A167865, A317099, A317100, A317712, A320222, A320226, A320269. Sequence in context: A275548 A260710 A093830 * A118033 A048810 A331680 Adjacent sequences:  A320265 A320266 A320267 * A320269 A320270 A320271 KEYWORD nonn AUTHOR Gus Wiseman, Oct 08 2018 STATUS approved

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Last modified April 3 19:43 EDT 2020. Contains 333198 sequences. (Running on oeis4.)