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A003239 Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.
(Formerly M1222)
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 (list; graph; refs; listen; history; text; internal format)



Also number of necklaces with 2n 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 n-compositions 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 n-composition of n.) a(n)=number of n-multisets in Z mod n whose sum is 0. - David Callan, Nov 05 2003


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).


Seiichi Manyama, Table of n, a(n) for n = 0..1669 (terms 0..200 from T. D. Noe)

R. Brualdi and M. Newman, An enumeration problem for a congruence equation, J. Res. Nat. Bureau Standards, B74 (1970), 37-40.

Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.

A. Elashvili, M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233--238. MR1691428 (2000c:13006).

A. Elashvili, M. Jibladze, D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173--188. MR1719140 (2000j:05009). See p. 174. - N. J. A. Sloane, Aug 06 2014

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 761

J. Malenfant, On the Matrix-Element Expansion of a Circulant Determinant, arXiv preprint arXiv:1502.06012 [math.NT], 2015.

J. Sawada, Generating rooted and free plane trees, ACM Transactions on Algorithms, Vol. 2 No. 1 (2006), 1-13.

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, vol. 19, No. 2 (1972), 200-204. - 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


a(n) = Sum_{d|n} (phi(n/d)*binomial(2d, d))/(2n), n>0.

a(n) = (1/n) Sum_{d|n} (phi(n/d)*binomial(2d-1, d)), n>0.

a(n) = A047996(2*n,n). - Philippe Deléham, Jul 25 2006

a(n) ~ 2^(2*n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015


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)];


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



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*d-1, d)) / n ); */

/* Joerg Arndt, Oct 21 2012 */


Cf. A002995, A057510, A000108, A022553, A084575, A037306.

Cf. A082936.

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




N. J. A. Sloane


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

Additional comments from Michael Somos

Elashvili et al. references supplied by Vladimir Popov, May 17 2014



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Last modified July 23 09:01 EDT 2017. Contains 289686 sequences.