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%I #40 Jul 14 2021 11:37:30
%S 0,1,1,4,25,220,2485,34300,559405,10525900,224449225,5348843500,
%T 140880765025,4063875715900,127418482316125,4314607214417500,
%U 156920190449147125,6100643259005795500,252476539015516440625,11081983532721088487500,514215436341672155715625
%N Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.
%C 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.
%C This sequence also enumerates a certain class of Feynman diagrams; see the references, links, and crossrefs below.
%D J. Bedford, On Perturbative Field Theory and Twistor String Theory, Ph.D. Thesis, 2007, Queen Mary, University of London.
%D 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.
%D 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.
%D M. L. Mangano and S. J. Parke, Multi-parton amplitudes in gauge theories, Physics Reports, Volume 200, Issue 6, February 1991, pp. 301-367.
%H Alois P. Heinz, <a href="/A268163/b268163.txt">Table of n, a(n) for n = 0..400</a>.
%H J. Bedford, <a href="http://arxiv.org/abs/0709.3478">On Perturbative Field Theory and Twistor String Theory</a>, arXiv:0709.3478 [hep-th], 2007, (see page 23).
%H D. Carmi, <a href="http://arxiv.org/abs/1109.5161">TeV scale strings and scattering amplitudes at the LHC</a>, arXiv:1109.5161 [hep-th], 2011-2015, (see page 23).
%H B. Feng, M. Luo, <a href="http://arxiv.org/abs/1111.5759">An introduction to on-shell recursion relations</a>, arXiv:1111.5759 [hep-th], 2011-2012, (see page 2).
%H K. Kampf, <a href="http://www.lupm.univ-montp2.fr/users/qcd/qcd14/qcd14_talks/tuesday/kampf.pdf">A new look at the nonlinear sigma model</a> (see pages 7 and 13).
%H M. L. Mangano, S. J. Parke, <a href="http://arxiv.org/abs/hep-th/0509223">Multi-parton amplitudes in gauge theories</a>, arXiv:hep-th/0509223, 2005, (see page 5).
%H G. Travaglini, <a href="http://www.umich.edu/~mctp/SciPrgPgs/events/2011/AMP2011/talks/Travaglini_Amplitudes_2011.pdf">Harmony of (super) form factors</a> (see page 3).
%H A. Volovich, <a href="http://www.slac.stanford.edu/econf/C0706044/pdf/03.pdf">Yang-Mills amplitudes and twistor string theory</a> (see bottom of page 1).
%H James Christopher Whitehead, <a href="http://etheses.dur.ac.uk/13949/">The Production of Pairs of Isolated Photons at Higher Orders in QCD</a>, Ph. D. Thesis, Durham University (UK, 2021).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</a>
%F 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
%F 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).
%F a(n) ~ sqrt(9-4*sqrt(3)) * ((12+9*sqrt(3))/11)^n * n^(n-1) / (3 * exp(n)). - _Vaclav Kotesovec_, Feb 24 2016
%e 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:
%e [[[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).
%p with(combinat):
%p b:= proc(n, i, v) option remember; `if`(n=0,
%p `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p add(multinomial(n, n-i*j, i$j)/j!*a(i)^j*
%p b(n-i*j, i-1, v-j), j=0..min(n/i, v))))
%p end:
%p a:= proc(n) option remember; `if`(n<2, n,
%p add(b(n, n+1-j, j), j=2..3))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 28 2016
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<3, [0, 1$2][n+1],
%p ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11)
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 28 2016
%t 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_ *)
%Y Cf. A001147. The number of labeled binary rooted non-planar trees.
%Y Cf. A025035. The number of labeled ternary rooted non-planar trees.
%Y Cf. A268172. The corresponding number of unlabelled trees.
%Y Cf. A005411. Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.
%Y 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).
%Y Cf. A005413. Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).
%Y Cf. A005414. Feynman diagrams of order 2n with vertex skeletons.
%Y Other sequences related to Feynman diagrams: A115974, A122023, A167872, A214298, A214299.
%Y Cf. A000311.
%K nonn
%O 0,4
%A _Murray R. Bremner_, Jan 27 2016
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