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.

%I A003239 M1222 #151 May 14 2025 13:49:50
%S A003239 1,1,2,4,10,26,80,246,810,2704,9252,32066,112720,400024,1432860,
%T A003239 5170604,18784170,68635478,252088496,930138522,3446167860,12815663844,
%U A003239 47820447028,178987624514,671825133648,2528212128776,9536895064400,36054433810102,136583761444364,518401146543812
%N A003239 Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.
%C A003239 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).
%C A003239 Also number of terms in polynomial expression for permanent of generic circulant matrix of order n.
%C A003239 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
%C A003239 a(n) is the number of cyclic equivalence classes of triangulations of a once-punctured n-gon. - _Esther Banaian_, May 06 2025
%D A003239 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 305 (see R(x)).
%D A003239 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973; page 80, Problem 3.13.
%D A003239 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A003239 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(b).
%H A003239 Seiichi Manyama, <a href="/A003239/b003239.txt">Table of n, a(n) for n = 0..1669</a> (terms 0..200 from T. D. Noe)
%H A003239 Michal Bassan, Serte Donderwinkel, and Brett Kolesnik, <a href="https://arxiv.org/abs/2406.05110">Graphical sequences and plane trees</a>, arXiv:2406.05110 [math.CO], 2024.
%H A003239 Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-5/Boman.pdf">Why do Fibonacci numbers appear in patterns of growth in nature?</a>, Fibonacci Quarterly, 55(5) (2017), 30-41.
%H A003239 R. Brualdi and M. Newman, <a href="http://nvlpubs.nist.gov/nistpubs/jres/74B/jresv74Bn1p37_A1b.pdf">An enumeration problem for a congruence equation</a>, J. Res. Nat. Bureau Standards, B74 (1970), 37-40.
%H A003239 CombOS - Combinatorial Object Server, <a href="http://combos.org/tree.html">Generate rooted plane trees</a>.
%H A003239 Paul Drube and Puttipong Pongtanapaisan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Drube/drube3.html">Annular Non-Crossing Matchings</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
%H A003239 A. Elashvili and M. Jibladze, <a href="http://dx.doi.org/10.1016/S0019-3577(98)80021-9">Hermite reciprocity for the regular representations of cyclic groups</a>, Indag. Math. (N.S.) 9(2) (1998), 233--238. MR1691428 (2000c:13006).
%H A003239 A. Elashvili, M. Jibladze, and D. Pataraia, <a href="http://dx.doi.org/10.1023/A:1018727630642">Combinatorics of necklaces and "Hermite reciprocity"</a>, J. Algebraic Combin. 10(2) (1999), 173--188. MR1719140 (2000j:05009). See p. 174. - _N. J. A. Sloane_, Aug 06 2014
%H A003239 M. L. Fredman, <a href="https://doi.org/10.1016/0097-3165(75)90008-4">A symmetry relationship for a class of partitions</a>, J. Combin. Theory Ser. A, 18 (1975), 199-202. See Eq. (4), a(n) = S(n,n,0).
%H A003239 F. Harary and R. W. Robinson, <a href="http://dx.doi.org/10.1515/crll.1975.278-279.322">The number of achiral trees</a>, J. Reine Angew. Math., 278 (1975), 322-335.
%H A003239 F. Harary and R. W. Robinson, <a href="/A002995/a002995_1.pdf">The number of achiral trees</a>, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
%H A003239 Thomas C. Hull and Tomohiro Tachi, <a href="https://arxiv.org/abs/1709.03210">Double-line rigid origami</a>, arXiv:1709.03210 [math.MG], 2017.
%H A003239 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=761">Encyclopedia of Combinatorial Structures 761</a>.
%H A003239 Benjamin Ruoyu Kan, <a href="https://dash.harvard.edu/handle/1/37376406">Polynomial Approximations for Quantum Hamiltonian Complexity</a>, Bachelor's thesis, Harvard Univ., 2023.
%H A003239 G. Labelle and P. Leroux, <a href="https://doi.org/10.1016/S0012-365X(96)83017-2">Enumeration of (uni- or bicolored) plane trees according to their degree distribution</a>, Disc. Math. 157 (1996), 227-240, Eq. (1.18).
%H A003239 J. Malenfant, <a href="http://arxiv.org/abs/1502.06012">On the Matrix-Element Expansion of a Circulant Determinant</a>, arXiv preprint arXiv:1502.06012 [math.NT], 2015.
%H A003239 Paul Melotti, Sanjay Ramassamy, and Paul Thévenin, <a href="https://arxiv.org/abs/2003.11006">Points and lines configurations for perpendicular bisectors of convex cyclic polygons</a>, arXiv:2003.11006 [math.CO], 2020.
%H A003239 J. Sawada, <a href="http://dx.doi.org/10.1145/1125994.1125995">Generating rooted and free plane trees</a>, ACM Transactions on Algorithms, 2(1) (2006), 1-13.
%H A003239 Hugh Thomas, <a href="http://arxiv.org/abs/math/0301048">The number of terms in the permanent and the determinant of a generic circulant matrix</a>, arXiv:math/0301048 [math.CO], 2003.
%H A003239 D. W. Walkup, <a href="http://dx.doi.org/10.1112/S0025579300005659">The number of plane trees</a>, Mathematika, 19(2) (1972), 200-204. - From _N. J. A. Sloane_, Jun 08 2012
%H A003239 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%H A003239 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A003239 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A003239 a(n) = Sum_{d|n} (phi(n/d)*binomial(2*d, d))/(2*n) for n > 0.
%F A003239 a(n) = (1/n)*Sum_{d|n} (phi(n/d)*binomial(2*d-1, d)) for n > 0.
%F A003239 a(n) = A047996(2*n, n). - _Philippe Deléham_, Jul 25 2006
%F A003239 a(n) ~ 2^(2*n-1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Aug 22 2015
%e A003239 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
%p A003239 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;
%p A003239 spec := [ C, {B=Union(Z,Prod(B,B)), C=Cycle(B)}, unlabeled ]; [seq(combstruct[count](spec, size=n), n=0..40)];
%t A003239 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 *)
%o A003239 (PARI)
%o A003239 C(n, k)=binomial(n,k);
%o A003239 a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(2*d, d)) / (2*n) );
%o A003239 /* or, second formula: */
%o A003239 /* a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(2*d-1,d)) / n ); */
%o A003239 /* _Joerg Arndt_, Oct 21 2012 */
%o A003239 (SageMath)
%o A003239 def A003239(n):
%o A003239     if n == 0: return 1
%o A003239     return sum(euler_phi(n/d)*binomial(2*d, d)/(2*n) for d in divisors(n))
%o A003239 print([A003239(n) for n in (0..29)]) # _Peter Luschny_, Dec 10 2020
%Y A003239 Cf. A002995, A057510, A000108, A022553, A082936, A084575, A037306.
%Y A003239 Column k=2 of A208183.
%Y A003239 Column k=1 of A261494.
%K A003239 nonn,nice,easy
%O A003239 0,3
%A A003239 _N. J. A. Sloane_
%E A003239 Sequence corrected and extended by Roderick J. Fletcher (yylee(AT)mail.ncku.edu.tw), Aug 1997
%E A003239 Additional comments from _Michael Somos_