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A003239
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Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.
(Formerly M1222)
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23
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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)=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 (callan(AT)stat.wisc.edu), Nov 05 2003
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REFERENCES
| R. Brualdi and M. Newman, An enumeration problem for a congruence equation, J. Res. Nat. Bureau Standards, B74 (1970), 37-40.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 761
Hugh Thomas, The number of terms in the permanent ..., arXiv:math.CO/0301048
Index entries for sequences related to necklaces
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
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FORMULA
| a(n) = sum {d|n} (phi(n/d)*C(2d, d))/(2n), n>0.
Or, equally, a(n) = (1/n) sum {d|n} (phi(n/d)*C(2d-1, d)), n>0.
a(n) = A047996(2*n,n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 25 2006
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MAPLE
| 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)];
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PROG
| (PARI) a(n)=if(n<1, n >= 0, sumdiv(n, k, eulerphi(n/k)*C(2*k, k))/(2*n)) where C(n, k)=if(k<0|k>n, 0, n!/k!/(n-k)!)
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CROSSREFS
| Cf. A002995, A057510, A000108, A022553, A084575.
Sequence in context: A148102 A179381 A096807 * A195924 A116673 A135410
Adjacent sequences: A003236 A003237 A003238 * A003240 A003241 A003242
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Sequence corrected and extended by Roderick J. Fletcher (yylee(AT)mail.ncku.edu.tw) 8/97. Additional comments from Michael Somos
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