A261494 -id:A261494 - 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

A082936 a(n) = (1/(3n))Sum_{d|n, d even} phi(2n/d)binomial(3d/2,d).

Original entry on oeis.org

1, 1, 3, 10, 43, 201, 1038, 5538, 30667, 173593, 1001603, 5864750, 34769374, 208267320, 1258579654, 7663720710, 46976034379, 289628805623, 1794932468571, 11175157356522, 69864075597643, 438403736549145, 2760351032959050, 17433869214973754, 110420300879752990

Offset: 0

N. J. A. Sloane, May 26 2003

nonn

a(n) = number of necklaces of n white beads and 2n black beads. - David Callan, Mar 28 2004

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141.
M. Isachenkov, I. Kirsch, V. Schomerus, Chiral Primaries in Strange Metals, arXiv preprint arXiv:1403.6857 [hep-th], 2014.

Cf. A003239.

Column k=2 of A261494.

with(numtheory): f := proc(n) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 then if d mod 2 = 0 then t1 := t1+phi(n/d)*binomial(3*d/2,d) fi; fi; od; 2*t1/(3*n); end; # use with n even

a[n_] := DivisorSum[n, EulerPhi[n/#]*Binomial[3#, #]&]/(3n); a[0] = 1; Array[a, 30, 0] (* Jean-François Alcover, Dec 02 2015 *)

C(n, k)=binomial(n,k);
a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(3*d,d)) / (3*n) );
/* or, second formula: */
/* a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(3*d-1,d)) / (2*n) ); */
/* Joerg Arndt, Oct 21 2012 */

From Joerg Arndt, Oct 21 2012: (Start)
a(n) = sum( d divides n, phi(n/d) * C(3*d,d) ) / (3*n) for n>=1, a(0)=1.
a(n) = sum( d divides n, phi(n/d) * C(3*d-1,d) ) / (2*n) for n>=1, a(0)=1.
a(n) = A047996(3*n,n).
(End)
a(n) ~ 3^(3*n) / (2^(2*n+1) * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

a(0)=1 prepended by Joerg Arndt, Oct 21 2012

A261497 Number of necklaces with n white beads and 3*n black beads.

Original entry on oeis.org

1, 1, 4, 19, 116, 776, 5620, 42288, 328756, 2615104, 21191904, 174303163, 1451430692, 12211799224, 103655949072, 886568153744, 7633233556276, 66105170315084, 575445692499952, 5032380942945322, 44191451788248416, 389514699013012242, 3444925385336301684

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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

Column k=3 of A261494.

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(4*n/d, n/d)
             *phi(d), d=divisors(n))/(4*n)):
seq(a(n), n=0..25);

a(n) = 1/(4*n) * Sum_{d|n} C(4*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 2^(8*n-3/2) / (3^(3*n) * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

A261498 Number of necklaces with n white beads and 4*n black beads.

Original entry on oeis.org

1, 1, 5, 31, 245, 2126, 19811, 192130, 1922741, 19692535, 205446630, 2175519380, 23322657491, 252631900236, 2760768051914, 30400169157656, 336977765092789, 3757141504436393, 42107201595510563, 474084628585822413, 5359833704140820870, 60823006052351729266

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..920
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

Column k=4 of A261494.

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(5*n/d, n/d)
             *phi(d), d=divisors(n))/(5*n)):
seq(a(n), n=0..25);

a(n) = 1/(5*n) * Sum_{d|n} C(5*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 5^(5*n-1/2) / (sqrt(Pi) * 2^(8*n+3/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

A261499 Number of necklaces with n white beads and 5*n black beads.

Original entry on oeis.org

1, 1, 6, 46, 446, 4751, 54132, 642342, 7861662, 98480332, 1256569506, 16273981758, 213378976004, 2826867619109, 37782553160820, 508840821830546, 6898459216311582, 94070535317459018, 1289430373206452136, 17755914760643605782, 245518560760433583946

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..850
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

Column k=5 of A261494.

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(6*n/d, n/d)
             *phi(d), d=divisors(n))/(6*n)):
seq(a(n), n=0..25);

a(n) = 1/(6*n) * Sum_{d|n} C(6*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 6^(6*n) / (2 * sqrt(3*Pi) * 5^(5*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

A261500 Number of necklaces with n white beads and 6*n black beads.

Original entry on oeis.org

1, 1, 7, 64, 735, 9276, 124936, 1753074, 25366335, 375677659, 5667212132, 86775157140, 1345153548264, 21069043965984, 332927800269694, 5301031234085664, 84967018635587775, 1369846562874360887, 22199151536133457885, 361411377745122110422, 5908312923795257331460

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..800
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

Column k=6 of A261494.

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(7*n/d, n/d)
             *phi(d), d=divisors(n))/(7*n)):
seq(a(n), n=0..25);

a(n) = 1/(7*n) * Sum_{d|n} C(7*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 7^(7*n-1/2) / (2 * sqrt(3*Pi) * 6^(6*n) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

A261501 Number of necklaces with n white beads and 7*n black beads.

Original entry on oeis.org

1, 1, 8, 85, 1128, 16451, 255704, 4141383, 69159400, 1182125128, 20581159608, 363704640476, 6506965279992, 117626432708864, 2145180358634664, 39421026305282660, 729242353169440744, 13568988503585900648, 253785064586356459616, 4768543107831461199897

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..760
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

Column k=7 of A261494.

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(8*n/d, n/d)
             *phi(d), d=divisors(n))/(8*n)):
seq(a(n), n=0..25);

a(n) = 1/(8*n) * Sum_{d|n} C(8*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 2^(24*n-2) / (sqrt(Pi) * 7^(7*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

A261495 Number of necklaces with n white beads and n^2 black beads.

Original entry on oeis.org

1, 1, 3, 19, 245, 4751, 124936, 4141383, 166237161, 7847250409, 426342182761, 26219808548110, 1801378010581175, 136784412621194274, 11378390032696241010, 1029218687419565103111, 100592759623604055645649, 10565465772302876757883823, 1186893721789951847976898669

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..335
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

Main diagonal of A261494.

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial((n^2+n)/d, n/d)
             *phi(d), d=divisors(n))/(n^2+n)):
seq(a(n), n=0..20);

a[n_] := If[n==0, 1, DivisorSum[n, Binomial[(n^2+n)/#, n/#]*EulerPhi[#]&]/ (n^2 + n)];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 25 2017, translated from Maple *)

a(n) = if(n<1, 1, sumdiv(n, d, binomial((n^2 + n)/d, n/d) * eulerphi(d)) / (n^2 + n));
for(n=0, 20, print1(a(n),", ")) \\ Indranil Ghosh, Mar 25 2017

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

a(n) ~ exp(n+1/2) * n^(n-5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 22 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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A082936 a(n) = (1/(3*n))*Sum_{d|n, d even} phi(2*n/d)*binomial(3d/2,d).

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A261497 Number of necklaces with n white beads and 3*n black beads.

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A261498 Number of necklaces with n white beads and 4*n black beads.

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A261499 Number of necklaces with n white beads and 5*n black beads.

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Maple

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A261500 Number of necklaces with n white beads and 6*n black beads.

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Programs

Maple

Formula

A261501 Number of necklaces with n white beads and 7*n black beads.

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Programs

Maple

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A261495 Number of necklaces with n white beads and n^2 black beads.

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A082936 a(n) = (1/(3n))Sum_{d|n, d even} phi(2n/d)binomial(3d/2,d).