A208183 -id:A208183 - 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

A137729 Number of circular permutations of the multiset {1,1,2,2,...,n,n} (up to rotations).

Original entry on oeis.org

1, 1, 2, 16, 318, 11352, 623760, 48648960, 5108105520, 694702028160, 118794043549440, 24946749109094400, 6311527524161798400, 1893458257242791500800, 664603848292138865510400, 269829162406607158901145600

Offset: 0

Max Alekseyev, Feb 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A114938, A137730, A137737, A137749.

Row n=2 of A208183.

a(n) = ((2*n)!/2^n + n!) / (2*n).

a(0)=1 prepended by Alois P. Heinz, Jul 28 2015

A118644 Number of distinct (n red, n blue, n green)-bead necklaces.

Original entry on oeis.org

1, 2, 16, 188, 2896, 50452, 953056, 19003476, 394397776, 8439756848, 185033251616, 4137181680700, 94020327215200, 2166105078791448, 50489825388328608, 1188777328563914488, 28236363841989180496, 675879582290807439796, 16289254212704277185152

Offset: 0

Jacob A. Siehler, May 10 2006

Alois P. Heinz, Table of n, a(n) for n = 0..700

Cf. A003239.

Column k=3 of A208183.

f[n_] := Plus @@ (EulerPhi[n/# ]Binomial[3#, # ]Binomial[2#, # ] & /@ Divisors[n])/(3n)

Sum_{d|n} ( phi(n/d)(3d Choose d)(2d Choose d) )/3n for n>0. [Corrected by several correspondents, Mar 04 2010]

a(n) ~ 3^(3*n-1/2) / (2*Pi*n^2). - Vaclav Kotesovec, Aug 23 2015

A207816 Number of distinct necklaces with n red, n green, n blue and n white beads.

Original entry on oeis.org

1, 6, 318, 30804, 3941598, 586637256, 96197661156, 16875655269948, 3111284141045598, 595909785174057204, 117634021777132574568, 23797087019979071174580, 4912693780461352534397604, 1031629572413246016139181544, 219809927417367534490107035244, 47426945434432859336092700072304

Offset: 0

Thotsaporn Thanatipanonda, Feb 20 2012

nonn

			For n=1, a(1)=6 since for four beads necklaces with each bead from each of the four colors say (R,G,B,W), we can arrange as following, [R,G,B,W], [R,G,W,B], [R,B,G,W], [R,B,W,G], [R,W,G,B] and [R,W,B,G].

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A003239, A118644, A137729.

Column k=4 of A208183. - Alois P. Heinz, Feb 24 2012

with(combinat): with(numtheory):
# This formula comes from Polya Counting Theorem:
# Z(C_n) = add(phi(d)*(a_d)^(n/d), d in divisors(n))/n;
PolyaBrace:= proc(S) option remember; local n, s, d;
               n:= add(s, s=S);
               add(phi(d) *PolyaCoeff(d, S), d=divisors(n))/n
             end:
# Find coeff of prod(a[i]^s[i], i=1..n) of a_d^(n/d) (symmetric function)
PolyaCoeff:= proc(d, S) option remember; local n, pow, s;
               n:= add(s, s=S);
               pow:= n/d;
               if {seq(s mod d, s = S)} = {0}
                  then multinomial(pow, seq(s/d, s = S))
                  else 0
               fi:
             end:
a:= n-> `if`(n=0, 1, PolyaBrace([n$4])):
seq(a(n), n=0..20);

a[n_] := DivisorSum[n, EulerPhi[n/#] (4#)!/(#!^4 * 4n)&]; a[0]=1;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)

a(n) = Sum_{d|n} phi(n/d)*(4*d)!/(d!^4*4*n) if n>0 and a(0) = 1. - Alois P. Heinz, Feb 24 2012

a(n) ~ 2^(8*n-5/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Aug 23 2015

A208184 Number of distinct n-colored necklaces with 3 beads per color.

Original entry on oeis.org

1, 1, 4, 188, 30804, 11211216, 7623616080, 8690922240480, 15391623287043360, 40018220546304026880, 146226577876194816241920, 725283826265926287362419200, 4746982642910487550771226611200, 40045545575592872978305843519334400

Offset: 0

Alois P. Heinz, Feb 24 2012

nonn

			a(0) = 1: the empty necklace.
a(1) = 1: {000}.
a(2) = 4: {000111, 001011, 010011, 010101}.

Alois P. Heinz, Table of n, a(n) for n = 0..80

Row n=3 of A208183.

Cf. A000010, A000142.

with(numtheory);
a:= n-> `if`(n=0, 1, add(phi(3/d) *(n*d)!/(d!^n *3*n), d={1, 3})):
seq(a(n), n=0..20);

Flatten[{1, Table[(3*n)!/(3*n*6^n) + 2*(n-1)!/3, {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 23 2015 *)

a(n) = Sum_{d|3} phi(3/d)*(n*d)!/(d!^n*n*3) if n>0 and a(0) = 1.

For n > 0, a(n) = (3*n)!/(3*n*6^n) + 2*(n-1)!/3. - Vaclav Kotesovec, Aug 23 2015

A208185 Number of distinct n-colored necklaces with 4 beads per color.

Original entry on oeis.org

1, 1, 10, 2896, 3941598, 15277017432, 135277939358160, 2374127830286012160, 74701932179186551241520, 3911393168902074440088524160, 321715999535364496261149134365440, 39702971502659332476270701578180454400, 7081620512071831837127802029303335215878400

Offset: 0

Alois P. Heinz, Feb 24 2012

nonn

			a(0) =  1: the empty necklace.
a(1) =  1: {0000}.
a(2) = 10: {00001111, 00010111, 00100111, 01000111, 00011011, 00110011, 00101011, 01010011, 01001011, 01010101}.

Alois P. Heinz, Table of n, a(n) for n = 0..50

Row n=4 of A208183.

Cf. A000010, A000142.

with(numtheory);
a:= n-> `if`(n=0, 1, add(phi(4/d) *(n*d)!/(d!^n *4*n), d={1,2,4})):
seq(a(n), n=0..15);

Flatten[{1, Table[(4*n)!/(4*n*24^n) + (n-1)!/2 + (2*n-1)!/2^(n+1), {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 23 2015 *)

a(n) = Sum_{d|4} phi(4/d)*(n*d)!/(d!^n*n*4) if n>0 and a(0) = 1.

For n > 0, a(n) = (4*n)!/(4*n*24^n) + (n-1)!/2 + (2*n-1)!/2^(n+1). - Vaclav Kotesovec, Aug 23 2015

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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A137729 Number of circular permutations of the multiset {1,1,2,2,...,n,n} (up to rotations).

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A118644 Number of distinct (n red, n blue, n green)-bead necklaces.

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A207816 Number of distinct necklaces with n red, n green, n blue and n white beads.

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A208184 Number of distinct n-colored necklaces with 3 beads per color.

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A208185 Number of distinct n-colored necklaces with 4 beads per color.

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A208186 Number of distinct n-colored necklaces with 5 beads per color.

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A208187 Number of distinct n-colored necklaces with 6 beads per color.

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