

A003239


Number of rooted planar trees with n nonroot nodes: circularly cycling the subtrees at the root gives equivalent trees.
(Formerly M1222)


35



1, 1, 2, 4, 10, 26, 80, 246, 810, 2704, 9252, 32066, 112720, 400024, 1432860, 5170604, 18784170, 68635478, 252088496, 930138522, 3446167860, 12815663844, 47820447028, 178987624514, 671825133648, 2528212128776, 9536895064400, 36054433810102, 136583761444364, 518401146543812
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OFFSET

0,3


COMMENTS

Also number of necklaces with 2*n beads, n white and n black (to get the correspondence, start at root, walk around outside of tree, use white if move away from the root, black if towards root).
Also number of terms in polynomial expression for permanent of generic circulant matrix of order n.
a(n) is the number of equivalence classes of ncompositions of n under cyclic rotation. (Given a necklace, split it into runs of white followed by a black bead and record the lengths of the white runs. This gives an ncomposition of n.) a(n) is the number of nmultisets in Z mod n whose sum is 0.  David Callan, Nov 05 2003


REFERENCES

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 305 (see R(x)).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973; page 80, Problem 3.13.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(b).


LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1669 (terms 0..200 from T. D. Noe)
Bruce M. Boman, ThienNam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, Fibonacci Quarterly, 55(5) (2017), 3041.
R. Brualdi and M. Newman, An enumeration problem for a congruence equation, J. Res. Nat. Bureau Standards, B74 (1970), 3740.
CombOS  Combinatorial Object Server, Generate rooted plane trees.
Paul Drube and Puttipong Pongtanapaisan, Annular NonCrossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9(2) (1998), 233238. MR1691428 (2000c:13006).
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10(2) (1999), 173188. MR1719140 (2000j:05009). See p. 174.  N. J. A. Sloane, Aug 06 2014
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combin. Theory Ser. A, 18 (1975), 199202. See Eq. (4), a(n) = S(n,n,0).
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322335.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322335. (Annotated scanned copy)
Thomas C. Hull and Tomohiro Tachi, Doubleline rigid origami, arXiv:1709.03210 [math.MG], 2017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 761.
G. Labelle, P. Leroux, Enumeration of (uni or bicolored) plane trees according to their degree distribution, Disc. Math. 157 (1996), 227240, Eq. (1.18).
J. Malenfant, On the MatrixElement Expansion of a Circulant Determinant, arXiv preprint arXiv:1502.06012 [math.NT], 2015.
Paul Melotti, Sanjay Ramassamy, Paul Thévenin, Points and lines configurations for perpendicular bisectors of convex cyclic polygons, arXiv:2003.11006 [math.CO], 2020.
J. Sawada, Generating rooted and free plane trees, ACM Transactions on Algorithms, 2(1) (2006), 113.
Hugh Thomas, The number of terms in the permanent and the determinant of a generic circulant matrix, arXiv:math/0301048 [math.CO], 2003.
D. W. Walkup, The number of plane trees, Mathematika, 19(2) (1972), 200204.  From N. J. A. Sloane, Jun 08 2012
Index entries for sequences related to necklaces
Index entries for sequences related to rooted trees
Index entries for sequences related to trees


FORMULA

a(n) = Sum_{dn} (phi(n/d)*binomial(2*d, d))/(2*n) for n > 0.
a(n) = (1/n)*Sum_{dn} (phi(n/d)*binomial(2*d1, d)) for n > 0.
a(n) = A047996(2*n, n).  Philippe Deléham, Jul 25 2006
a(n) ~ 2^(2*n1) / (sqrt(Pi) * n^(3/2)).  Vaclav Kotesovec, Aug 22 2015


MAPLE

with(numtheory): A003239 := proc(n) local t1, t2, d; t2 := divisors(n); t1 := 0; for d in t2 do t1 := t1+phi(n/d)*binomial(2*d, d)/(2*n); od; t1; end;
spec := [ C, {B=Union(Z, Prod(B, B)), C=Cycle(B)}, unlabeled ]; [seq(combstruct[count](spec, size=n), n=0..40)];


MATHEMATICA

a[n_] := Sum[ EulerPhi[n/k]*Binomial[2k, k]/(2n), {k, Divisors[n]}]; a[0] = 1; Table[a[n], {n, 0, 25}] (* JeanFrançois Alcover, Apr 11 2012 *)


PROG

(PARI)
C(n, k)=binomial(n, k);
a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(2*d, d)) / (2*n) );
/* or, second formula: */
/* a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(2*d1, d)) / n ); */
/* Joerg Arndt, Oct 21 2012 */
(SageMath)
def A003239(n):
if n == 0: return 1
return sum(euler_phi(n/d)*binomial(2*d, d)/(2*n) for d in divisors(n))
print([A003239(n) for n in (0..29)]) # Peter Luschny, Dec 10 2020


CROSSREFS

Cf. A002995, A057510, A000108, A022553, A082936, A084575, A037306.
Column k=2 of A208183.
Column k=1 of A261494.
Sequence in context: A148102 A179381 A096807 * A195924 A116673 A135410
Adjacent sequences: A003236 A003237 A003238 * A003240 A003241 A003242


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Sequence corrected and extended by Roderick J. Fletcher (yylee(AT)mail.ncku.edu.tw), Aug 1997
Additional comments from Michael Somos


STATUS

approved



