login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves. 2
0, 1, 1, 4, 25, 220, 2485, 34300, 559405, 10525900, 224449225, 5348843500, 140880765025, 4063875715900, 127418482316125, 4314607214417500, 156920190449147125, 6100643259005795500, 252476539015516440625, 11081983532721088487500, 514215436341672155715625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This can also be interpreted as the number of multilinear monomials of degree n in a nonassociative algebra with an (anti)commutative binary operation and a completely (skew-)symmetric ternary operation; the number of variables in the monomial corresponds to the number of leaves in the tree.

This sequence also enumerates a certain class of Feynman diagrams; see the references, links, and crossrefs below.

REFERENCES

J. Bedford, On Perturbative Field Theory and Twistor String Theory, Ph.D. Thesis, 2007, Queen Mary, University of London.

B. Feng and M. Luo, An introduction to on-shell recursion relations, Review Article, Frontiers of Physics, October 2012, Volume 7, Issue 5, pp. 533-575.

K. Kampf, A new look at the nonlinear sigma model, 17th International Conference in Quantum Chromodynamics (QCD 14), Nuclear and Particle Physics Proceedings, Volumes 258-259, January-February 2015, pp. 86-89.

M. L. Mangano and S. J. Parke, Multi-parton amplitudes in gauge theories, Physics Reports, Volume 200, Issue 6, February 1991, pp. 301-367.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400.

J. Bedford, On Perturbative Field Theory and Twistor String Theory, arXiv:0709.3478 [hep-th], 2007, (see page 23).

D. Carmi, TeV scale strings and scattering amplitudes at the LHC, arXiv:1109.5161 [hep-th], 2011-2015, (see page 23).

B. Feng, M. Luo, An introduction to on-shell recursion relations, arXiv:1111.5759 [hep-th], 2011-2012, (see page 2).

K. Kampf, A new look at the nonlinear sigma model (see pages 7 and 13).

M. L. Mangano, S. J. Parke, Multi-parton amplitudes in gauge theories, arXiv:hep-th/0509223, 2005, (see page 5).

G. Travaglini, Harmony of (super) form factors (see page 3).

A. Volovich, Yang-Mills amplitudes and twistor string theory (see bottom of page 1).

Wikipedia, Feynman diagram

FORMULA

a(n) = ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11 for n>2. - Alois P. Heinz, Jan 28 2016

Because of Koszul duality for operads, the exponential generating function is the compositional inverse of the power series x-x^2/2-x^3/6 (email of Vladimir Dotsenko to Murray R. Bremner, Jan 28 2016).

a(n) ~ sqrt(9-4*sqrt(3)) * ((12+9*sqrt(3))/11)^n * n^(n-1) / (3 * exp(n)). - Vaclav Kotesovec, Feb 24 2016

EXAMPLE

For n = 4 and using the monomial interpretation, the 25 multilinear monomials of degree 4 are as follows, where [-,-] is the binary operation and (-,-,-) is the ternary operation:

[[[a,b],c],d], [[[a,b],d],c], [[[a,c],b],d], [[[a,c],d],b], [[[a,d],b],c], [[[a,d],c],b], [[[b,c],a],d], [[[b,c],d],a], [[[b,d],a],c], [[[b,d],c],a], [[[c,d],a],b], [[[c,d],b],a], [[a,b],[c,d]], [[a,c],[b,d]], [[a,d],[b,c]], [(a,b,c),d], [(a,b,d),c], [(a,c,d),b], [(b,c,d),a], ([a,b],c,d), ([a,c],b,d), ([a,d],b,c), ([b,c],a,d), ([b,d],a,c), ([c,d],a,b).

MAPLE

with(combinat):

b:= proc(n, i, v) option remember; `if`(n=0,

      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,

       add(multinomial(n, n-i*j, i$j)/j!*a(i)^j*

       b(n-i*j, i-1, v-j), j=0..min(n/i, v))))

    end:

a:= proc(n) option remember; `if`(n<2, n,

      add(b(n, n+1-j, j), j=2..3))

    end:

seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2016

# second Maple program:

a:= proc(n) option remember; `if`(n<3, [0, 1$2][n+1],

       ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11)

    end:

seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2016

MATHEMATICA

a[0]=0; a[1]=1; a[2]=1; a[n_]:=a[n]=(12(2n-3)a[n-1]+(3n-5)(3n-7)a[n-2])/11; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A001147. The number of labeled binary rooted non-planar trees.

Cf. A025035. The number of labeled ternary rooted non-planar trees.

Cf. A268172. The corresponding number of unlabelled trees.

Cf. A005411. Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.

Cf. A005412. Number of non-vanishing Feynman diagrams of order 2n for the vacuum polarization (the proper two-point function of the photon) and for the self-energy (the proper two-point function of the electron) in quantum electrodynamics (QED).

Cf. A005413. Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).

Cf. A005414. Feynman diagrams of order 2n with vertex skeletons.

Other sequences related to Feynman diagrams: A115974, A122023, A167872, A214298, A214299.

Cf. A000311.

Sequence in context: A209319 A273996 A246530 * A050386 A246524 A246528

Adjacent sequences:  A268160 A268161 A268162 * A268164 A268165 A268166

KEYWORD

nonn

AUTHOR

Murray R. Bremner, Jan 27 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 23:02 EDT 2019. Contains 321343 sequences. (Running on oeis4.)