A167872 -id:A167872 - cp's OEIS frontend

A200659 Triangle T(n,k), read by rows, given by (1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 71, 132, 76, 16, 1, 461, 955, 670, 200, 25, 1, 3447, 7782, 6309, 2374, 435, 36, 1, 29093, 70441, 63833, 28413, 6713, 833, 49, 1, 273343, 701352, 694500, 351512, 99868, 16240, 1456, 64, 1

Offset: 0

Philippe Deléham, Nov 20 2011

			Triangle begins :
1
1, 1
3, 4, 1
13, 21, 9, 1
71, 132, 76, 16, 1
461, 955, 670, 200, 25, 1
3447, 7782, 6309, 2374, 435, 36, 1
29093, 70441, 63833, 28413, 6713, 833, 49, 1
273343, 701352, 694500, 351512, 99868, 16240, 1456, 64, 1

Cf. A111537, A111546, A111556, A167872

Sum_{k, 0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A003319(n), A111537(n), A111546(n), A111556(n), A177354(n-1) for x = -2,-1,0,1,2,3,4 respectively.
Sum_ {k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A111537(n), A167872(n) for x = 0,1,2 respectively.
T(k+1,k)=(k+1)^2.

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

Original entry on oeis.org

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

A321963 Stieltjes generated from the sequence m, m+1, m+2, m+3, .... where m = 4.

Original entry on oeis.org

1, 4, 36, 444, 6636, 114084, 2194596, 46460124, 1070653356, 26650132164, 712373143716, 20355134459004, 619356569885676, 20002325474150244, 683641504802995236, 24662695086736585884, 936845038595867508396, 37388655553571504769924, 1564425694139017014501156

Offset: 0

Peter Luschny, Dec 26 2018

nonn

See A321964 for the definitions.

A000007 (m=0), A001147 (m=1), A000698 (m=2), A167872 (m=3), this sequence (m=4).

a(n) = A127059(n)/3.

A321963List := proc(len) local S, k, m, cf, ser;
    S := [seq(k+4, k = 0..len)]: m := 1;
    for k from len by -1 to 1 do
        m := 1 - S[k]*x/m od;
    cf := 1/m:
    ser := series(cf, x, len);
    seq(coeff(ser, x, n), n = 0..len-1) end:
A321963List(19);

T[n_, k_] := T[n, k] = If[k == n, n + 1, Sum[T[j + k, k] T[n - j, k + 1], {j, 0, n - k - 1}]]; a[n_] := T[n + 2, 2]/3; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 22 2019, from A127059 *)

a(n) ~ 2^(n + 5/2) * n^(n+3) / (3*exp(n)). - Vaclav Kotesovec, Jan 02 2019

A226270 Number of primitive permutations with n buds and 3 red or blue elements.

Original entry on oeis.org

1, 7, 69, 843, 12081, 197127

Offset: 0

N. J. A. Sloane, Jun 05 2013

Adrian Ocneanu, On the inner structure of a permutation: bicolored partitions and Eulerians, trees and primitives; arXiv preprint arXiv:1304.1263 [math.CO], 2013.
Jianghao Xu and Di Yang, Galilean symmetry of the KdV hierarchy, arXiv:2403.04631 [math-ph], 2024. See p. 6.

Cf. A000698, A001147, A167872, A226297, A226298, A226299.

Conjectural recurrence: a(n) = (1/3)*(n+1)*d(n+2) - Sum_{k = 0..n-1} d(n+1-k)*a(k)  with a(0)= 1, where d(n) = (2*n-1)!! = A001147(n). If true then the sequence continues [1, 7, 69, 843, 12081, 197127, 3595509, 72393003, 1594224801, 38123341767, 984142175589, ...]. - Peter Bala, Dec 26 2019
From Peter Bala, Jul 07 2022: (Start)
Let A(x) = 1 + 7*x + 69*x^2 + 843*x^3 + 12081*x^4 + 197127*x^5 + ... denote the o.g.f. Then it appears that 1 + 2*x*A(x) = 1/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - 9*x/(1 - 8*x/(1 - ...)))))))), a continued fraction of Stieltjes type, and also that 1 + 2*x*A(x) =  1/(1 + 3*x - 5*x/(1 + 5*x - 7*x/ (1 + 7*x - 9*x/ (1 + 9*x - 11*x/ (1 + 11*x - ...))))).
If true, then 1 + x/(1 - 2*x - 6*x^2*A(x)) = 1 + x + 2*x^2 + 10*x^3 + 74*x^4 + 706*x^5 + ... is the g.f. of A000698. (End)

cp's OEIS Frontend

A200659 Triangle T(n,k), read by rows, given by (1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

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

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A321963 Stieltjes generated from the sequence m, m+1, m+2, m+3, .... where m = 4.

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Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

A226270 Number of primitive permutations with n buds and 3 red or blue elements.

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Author

Keywords

Links

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Formula