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

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

0,3

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).

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

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

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