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 A000107 Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates. (Formerly M1442 N0570) 23

%I M1442 N0570

%S 0,1,2,5,13,35,95,262,727,2033,5714,16136,45733,130046,370803,1059838,

%T 3035591,8710736,25036934,72069134,207727501,599461094,1731818878,

%U 5008149658,14496034714,41993925955,121747732406,353221737526,1025471857282,2978995353959,8658997820084

%N Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 61, 62 (2.1.8-2.1.10)

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000107/b000107.txt">Table of n, a(n) for n=0..200</a>

%H Bernhard Gittenberger, Emma Yu Jin, Michael Wallner, <a href="https://arxiv.org/abs/1707.02144">On the shape of random Pólya structures</a>, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.

%H R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

%H R. Harary, R. W. Robinson, <a href="http://dx.doi.org/10.1007/BF02579217">Isomorphic factorizations VIII: bisectable trees</a>, Combinatorica 4 (2) (1984) 169-179, eq. (4.12).

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=123">Encyclopedia of Combinatorial Structures 123</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1603.00077">Topologically distinct sets of non-intersecting circles in the plane</a>, arXiv:1603.00077 [math.CO] (2016), Table 6.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F G.f.: A000081(x)/(1-A000081(x)), where A000081(x) is the g.f. of A000081 [Harary-Robinson]. - _R. J. Mathar_, Sep 16 2015

%p with(numtheory): b:= proc(n) option remember; `if`(n<2, n, add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1) /(n-1)) end: a:= proc(n) option remember; b(n) +add(a(n-i) *b(i), i=1..n-1) end: seq(a(n), n=0..26); # _Alois P. Heinz_, Jun 02 2009

%t b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[ Sum[ d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1); a[n_] := a[n] = b[n] + Sum[ a[n-i]*b[i], {i, 1, n-1}]; Table[ a[n], {n, 0, 26}](* _Jean-François Alcover_, Mar 07 2012, after _Alois P. Heinz_ *)

%Y INVERT transform of A000081. Cf. A000243, A000269, A000312, A000444, A000485, A000524-A000526.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E Better description from _Christian G. Bower_, Apr 15 1998

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Last modified June 17 05:19 EDT 2019. Contains 324183 sequences. (Running on oeis4.)