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 A300660 Number of unlabeled rooted phylogenetic trees with n (leaf-) nodes such that for each inner node all children are either leaves or roots of distinct subtrees. 37
 0, 1, 1, 2, 3, 6, 13, 30, 72, 182, 467, 1222, 3245, 8722, 23663, 64758, 178459, 494922, 1380105, 3867414, 10884821, 30756410, 87215419, 248117618, 707952902, 2025479210, 5809424605, 16700811214, 48113496645, 138884979562, 401645917999, 1163530868090 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS From Gus Wiseman, Jul 31 2018 and Feb 06 2020: (Start) a(n) is the number of lone-child-avoiding rooted identity trees whose leaves form an integer partition of n. For example, the following are the a(6) = 13 lone-child-avoiding rooted identity trees whose leaves form an integer partition of 6.   6,   (15),   (24),   (123), (1(23)), (2(13)), (3(12)),   (1(14)),   (1(1(13))),   (12(12)), (1(2(12))), (2(1(12))),   (1(1(1(12)))). (End) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..2079 Wikipedia, Phylogenetic tree FORMULA a(n) ~ c * d^n / n^(3/2), where d = 3.045141208159736483720243229947630323380565686... and c = 0.2004129296838557718008171812000512670126... - Vaclav Kotesovec, Aug 27 2018 EXAMPLE :   a(3) = 2:        :   a(4) = 3:                      : :      o       o     :        o         o        o      : :     / \     /|\    :       / \       / \     /( )\    : :    o   N   N N N   :      o   N     o   N   N N N N   : :   ( )              :     / \       /|\                : :   N N              :    o   N     N N N               : :                    :   ( )                            : :                    :   N N                            : From Gus Wiseman, Feb 06 2020: (Start) The a(2) = 1 through a(6) = 13 unlabeled rooted phylogenetic semi-identity trees:   (oo) (ooo)     (oooo)         (ooooo)             (oooooo)        ((o)(oo)) ((o)(ooo))     ((o)(oooo))         ((o)(ooooo))                  ((o)((o)(oo))) ((oo)(ooo))         ((oo)(oooo))                                 ((o)((o)(ooo)))     ((o)(oo)(ooo))                                 ((oo)((o)(oo)))     (((o)(oo))(ooo))                                 ((o)((o)((o)(oo)))) ((o)((o)(oooo)))                                                     ((o)((oo)(ooo)))                                                     ((oo)((o)(ooo)))                                                     ((o)(oo)((o)(oo)))                                                     ((o)((o)((o)(ooo))))                                                     ((o)((oo)((o)(oo))))                                                     ((oo)((o)((o)(oo))))                                                     ((o)((o)((o)((o)(oo))))) (End) MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(b(n-i*j, i-1)*binomial(a(i), j), j=0..n/i)))     end: a:= n-> `if`(n=0, 0, 1+b(n, n-1)): seq(a(n), n=0..30); MATHEMATICA b[0, _] = 1; b[_, _?NonPositive] = 0; b[n_, i_] := b[n, i] = Sum[b[n-i*j, i-1]*Binomial[a[i], j], {j, 0, n/i}]; a[0] = 0; a[n_] := a[n] = 1 + b[n, n-1]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, May 03 2019, from Maple *) ursit[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]], UnsameQ@@#&], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}], n]; Table[Length[ursit[n]], {n, 10}] (* Gus Wiseman, Feb 06 2020 *) CROSSREFS Cf. A000081, A004111, A141268, A289501, A301467. Cf. A000669, A001678, A005804, A292504, A300660, A316653, A316654, A316656. The locally disjoint case is A316694. Cf. A276625, A306200, A319312, A331679, A331686, A331875. Sequence in context: A052937 A005554 A316766 * A077212 A076836 A296531 Adjacent sequences:  A300657 A300658 A300659 * A300661 A300662 A300663 KEYWORD nonn,eigen AUTHOR Alois P. Heinz, Jun 18 2018 STATUS approved

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Last modified April 17 02:26 EDT 2021. Contains 343059 sequences. (Running on oeis4.)