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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 September 30 21:27 EDT 2020. Contains 337440 sequences. (Running on oeis4.)