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A000107 Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.
(Formerly M1442 N0570)
20
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

T. D. Noe, Table of n, a(n) for n=0..200

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

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

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-Fran├žois Alcover, Mar 07 2012, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A057960 A227045 A007075 * A063028 A085810 A235611

Adjacent sequences:  A000104 A000105 A000106 * A000108 A000109 A000110

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Christian G. Bower, Apr 15 1998

STATUS

approved

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Last modified December 15 03:08 EST 2017. Contains 296020 sequences.