A000107 Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.
0, 1, 2, 5, 13, 35, 95, 262, 727, 2033, 5714, 16136, 45733, 130046, 370803, 1059838, 3035591, 8710736, 25036934, 72069134, 207727501, 599461094, 1731818878, 5008149658, 14496034714, 41993925955, 121747732406, 353221737526, 1025471857282, 2978995353959, 8658997820084
Offset: 0
References
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 61, 62 (2.1.8-2.1.10).
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
- Bernhard Gittenberger, Emma Yu Jin, Michael Wallner, On the shape of random Pólya structures, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.
- R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
- R. Harary, R. W. Robinson, Isomorphic factorizations VIII: bisectable trees, Combinatorica 4 (2) (1984) 169-179, eq. (4.12).
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 123
- R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 6.
- N. J. A. Sloane, Transforms
- Index entries for sequences related to rooted trees
- Index entries for sequences related to trees
Programs
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Maple
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
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Mathematica
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 *)
Formula
G.f.: A000081(x)/(1-A000081(x)), where A000081(x) is the g.f. of A000081 [Harary-Robinson]. - R. J. Mathar, Sep 16 2015
Extensions
Better description from Christian G. Bower, Apr 15 1998