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A046342 Number of 3-bead necklaces where each bead is a planted trivalent plane tree [or anything else enumerated by the Catalan numbers], by total number of nodes. 2
1, 1, 3, 8, 24, 74, 245, 815, 2796, 9707, 34186, 121562, 436298, 1577310, 5740299, 21008777, 77279892, 285544700, 1059332082, 3944254118, 14734260864, 55207053787, 207421476390, 781283558998, 2949675307082, 11160264942376, 42309912978708, 160700303600030 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

With offset = 3, a(n) is the number of forests having exactly three rooted plane trees with n total nodes. - Geoffrey Critzer, Feb 22 2013

LINKS

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

FORMULA

Plug g.f. for A000108, 1/2*(1-(1-4*x)^(1/2))/x, into cycle index for dihedral group D_6.

Cycle index for D_6: 1/6*Z[1]^3+1/2*Z[1]*Z[2]+1/3*Z[3].

a(n) = Sum_{j=0..3} A275431(n,j). - Alois P. Heinz, Sep 20 2017

MATHEMATICA

nn=30; Drop[CoefficientList[Series[ CycleIndex[SymmetricGroup[3], s]/.Table[s[i]->(1-(1-4x^i)^(1/2))/2, {i, 1, nn}], {x, 0, nn}], x], 3] (* Geoffrey Critzer, Feb 22 2013 *)

CROSSREFS

See A058855 (a 6-bead analog) for details.

Cf. A000108, A058855, A056711, A275431.

Sequence in context: A275856 A303607 A281872 * A238977 A182453 A000958

Adjacent sequences:  A046339 A046340 A046341 * A046343 A046344 A046345

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 19 2001

STATUS

approved

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Last modified February 15 22:28 EST 2019. Contains 320138 sequences. (Running on oeis4.)