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A000107 Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.
(Formerly M1442 N0570)
17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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).
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
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
a(n) ~ A340310 * A051491^n / sqrt(n). - Vaclav Kotesovec, Jan 04 2021
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
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 *)
CROSSREFS
Row sums of A339067.
INVERT transform of A000081.
Column k=1 of A008295.
Sequence in context: A339479 A227045 A007075 * A366088 A370841 A063028
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Better description from Christian G. Bower, Apr 15 1998
STATUS
approved

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Last modified March 19 07:25 EDT 2024. Contains 370955 sequences. (Running on oeis4.)