A005411 -id:A005411 - cp's OEIS frontend

A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

0, 1, 1, 4, 25, 220, 2485, 34300, 559405, 10525900, 224449225, 5348843500, 140880765025, 4063875715900, 127418482316125, 4314607214417500, 156920190449147125, 6100643259005795500, 252476539015516440625, 11081983532721088487500, 514215436341672155715625

Offset: 0

Murray R. Bremner, Jan 27 2016

nonn

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.

			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).

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.

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).
James Christopher Whitehead, The Production of Pairs of Isolated Photons at Higher Orders in QCD, Ph. D. Thesis, Durham University (UK, 2021).
Wikipedia, Feynman diagram

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.

with(combinat):
b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or nAlois 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

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 *)

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

A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Johannes W. Meijer, Jul 27 2016

nonn

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.
To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the inverse multinomial transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.
The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)

Richard P. Feynman, QED, The strange theory of light and matter, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.
Wikipedia, Feynman diagram

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.

A333114 Sum over all closed Deutsch paths of length n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 0, 1, 1, 5, 11, 44, 134, 529, 1902, 7793, 31068, 133641, 574259, 2594969, 11842726, 56083004, 269450143, 1333170844, 6703500545, 34548749471, 181026885253, 969167994094, 5273977173249, 29257773480987, 164894374634333, 945779302210358, 5507572390808676

Offset: 0

Alois P. Heinz, Mar 07 2020

nonn

Deutsch paths (named after their inventor Emeric Deutsch by Helmut Prodinger) are like Dyck paths where down steps can get to all lower levels. Open paths can end at any level, whereas closed paths have to return to the lowest level zero at the end.

			a(4) = (1/1)*(3/1) + 2/2 + 3/3 = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..800
Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020
Wikipedia, Counting lattice paths

Cf. A005043, A005411, A330169, A333098.

b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
      `if`(t and j<0, x/y, 1)*b(x-1, y+j, is(j>0)), j=[
      `if`(y=0, [][], -1), $1..x-1-y]))
    end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..30);

b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[If[t && j < 0, x/y, 1]* b[x-1, y+j, j > 0], {j, Join[If[y == 0, {}, {-1}], Range[x-1-y]]}]];
a[n_] := b[n, 0, False];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 19 2020, after Alois P. Heinz *)

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A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

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A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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A333114 Sum over all closed Deutsch paths of length n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p.

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A224978 The number of completed primitive graphs of n loops, in enumerating Feynman graphs in quantum field theory.

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