A082936 -id:A082936 - cp's OEIS frontend

A003239 Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.

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

Offset: 0

N. J. A. Sloane

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 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) is the number of n-multisets in Z mod n whose sum is 0. - David Callan, Nov 05 2003
a(n) is the number of cyclic equivalence classes of triangulations of a once-punctured n-gon. - Esther Banaian, May 06 2025

			As _David Callan_ said, a(n) is the number of n-multisets in Z mod n whose sum is 0. So for n = 4 the a(4)=10 multisets are (0, 0, 0, 0), (1, 1, 1, 1), (0, 1, 1, 2), (0, 0, 2, 2), (2, 2, 2, 2), (0, 0, 1, 3), (1, 2, 2, 3), (1, 1, 3, 3), (0, 2, 3, 3) and (3, 3, 3, 3). - _Boas Bakker_, Apr 21 2025

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).

Seiichi Manyama, Table of n, a(n) for n = 0..1669 (terms 0..200 from T. D. Noe)
Michal Bassan, Serte Donderwinkel, and Brett Kolesnik, Graphical sequences and plane trees, arXiv:2406.05110 [math.CO], 2024.
Bruce M. Boman, Thien-Nam 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), 30-41.
R. Brualdi and M. Newman, An enumeration problem for a congruence equation, J. Res. Nat. Bureau Standards, B74 (1970), 37-40.
CombOS - Combinatorial Object Server, Generate rooted plane trees.
Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing 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), 233--238. MR1691428 (2000c:13006).
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10(2) (1999), 173--188. 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), 199-202. 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), 322-335.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
Thomas C. Hull and Tomohiro Tachi, Double-line rigid origami, arXiv:1709.03210 [math.MG], 2017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 761.
Benjamin Ruoyu Kan, Polynomial Approximations for Quantum Hamiltonian Complexity, Bachelor's thesis, Harvard Univ., 2023.
G. Labelle and P. Leroux, Enumeration of (uni- or bicolored) plane trees according to their degree distribution, Disc. Math. 157 (1996), 227-240, Eq. (1.18).
J. Malenfant, On the Matrix-Element Expansion of a Circulant Determinant, arXiv preprint arXiv:1502.06012 [math.NT], 2015.
Paul Melotti, Sanjay Ramassamy, and 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), 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, 19(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

Cf. A002995, A057510, A000108, A022553, A082936, A084575, A037306.
Column k=2 of A208183.
Column k=1 of A261494.

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 */

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

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

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

Additional comments from Michael Somos

A261494 Number A(n,k) of necklaces with n white beads and k*n black beads; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 10, 1, 1, 1, 5, 19, 43, 26, 1, 1, 1, 6, 31, 116, 201, 80, 1, 1, 1, 7, 46, 245, 776, 1038, 246, 1, 1, 1, 8, 64, 446, 2126, 5620, 5538, 810, 1, 1, 1, 9, 85, 735, 4751, 19811, 42288, 30667, 2704, 1

Offset: 0

Alois P. Heinz, Aug 21 2015

For k>=1 is column k asymptotic to (k+1)^((k+1)*n-1/2) / (sqrt(2*Pi) * k^(k*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

			A(2,2) = 3: 000011, 000101, 001001.
A(3,2) = 10: 000000111, 000001011, 000010011, 000100011, 001000011, 010000011, 000010101, 000100101, 001000101, 001001001.
Square array A(n,k) begins:
  1,  1,    1,    1,     1,     1,      1, ...
  1,  1,    1,    1,     1,     1,      1, ...
  1,  2,    3,    4,     5,     6,      7, ...
  1,  4,   10,   19,    31,    46,     64, ...
  1, 10,   43,  116,   245,   446,    735, ...
  1, 26,  201,  776,  2126,  4751,   9276, ...
  1, 80, 1038, 5620, 19811, 54132, 124936, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Eric Weisstein's World of Mathematics, Necklace
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Columns k=0-10 give: A000012, A003239, A082936, A261497, A261498, A261499, A261500, A261501, A261502, A261503, A261504.
Main diagonal gives A261495.
Lower diagonal gives A261496.
Cf. A000010.

with(numtheory):
A:= (n, k)-> `if`(n=0, 1, add(binomial((k+1)*n/d, n/d)
                    *phi(d), d=divisors(n))/((k+1)*n)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := If[n==0, 1, DivisorSum[n, Binomial[(k+1)*n/#, n/#]*EulerPhi[#] /((k+1)*n)&]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

a(n,k) = if(n<1, 1, sumdiv(n, d, binomial((k + 1)*n/d, n/d) * eulerphi(d)) / ((k + 1)*n));
for(d=0, 14, for(n=0, d, print1(a(n, d - n),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

A(n,k) = 1/((k+1)*n) * Sum_{d|n} C((k+1)*n/d,n/d) * A000010(d) for n>0, A(0,k) = 1.

A(n,k) = 1/((k+1)*n)*Sum_{i=1..n} C((k+1)*gcd(n,i),gcd(n,i)) = 1/((k+1)*n)*Sum_{i=1..n} C((k+1)*n/gcd(n,i),n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)) for n >= 1, where phi = A000010. - Richard L. Ollerton, May 19 2021

cp's OEIS Frontend

A003239 Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.

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A261494 Number A(n,k) of necklaces with n white beads and k*n black beads; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A346577 a(n) = (1/(3n)) Sum_{d|n} mu(n/d) * binomial(3*d,d).

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cp's OEIS Frontend

A003239 Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.

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Maple

Mathematica

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A261494 Number A(n,k) of necklaces with n white beads and k*n black beads; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A346577 a(n) = (1/(3*n)) * Sum_{d|n} mu(n/d) * binomial(3*d,d).

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A346577 a(n) = (1/(3n)) Sum_{d|n} mu(n/d) * binomial(3*d,d).