cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2001

Keywords

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

Crossrefs

See A058855 (a 6-bead analog) for details.
Cf. A000108, A058855, A056711, A275431.

Programs

  • 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 *)

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

A056711 Plug g.f. for A000108 (minus the leading 1), 1/2*(1-(1-4*x)^(1/2))/x - 1, into cycle index for dihedral group D_3.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 28, 100, 358, 1309, 4772, 17556, 64782, 240090, 892662, 3329942, 12456782, 46725350, 175698056, 662193908, 2501118956, 9465771967, 35891640172, 136331485336, 518702002350, 1976588406300, 7543137149256, 28826327724850, 110304963059048
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2001

Keywords

Links

Crossrefs

Cf. A000108, A058855, A046342.
Column k=3 of A275431.

Formula

Cycle index for D_3: 1/6*Z[1]^3+1/2*Z[1]*Z[2]+1/3*Z[3].
O.g.f.: 1/6*A(x)^3 + 1/2*A(x)*A(x^2) + 1/3*A(x^3), with A(x):=1/2*(1-(1-4*x)^(1/2))/x - 1 (see the name). For the cycle index of the dihedral group D_n see A212355 for the Harary-Palmer reference, the formula and a link. - Wolfdieter Lang, Jun 02 2012

A059221 Number of 3-bead necklaces where each bead is an unlabeled rooted tree, by total number of nodes.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 43, 103, 255, 642, 1640, 4242, 11096, 29278, 77882, 208589, 562056, 1522606, 4144521, 11329608, 31090910, 85618848, 236530942, 655344579, 1820585313, 5070138554, 14151907583, 39584395796, 110938500821, 311481097178
Offset: 0

Views

Author

N. J. A. Sloane, Jan 18 2001

Keywords

Crossrefs

See A058855 (the 6-bead analog) for details.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.