A115974 -id:A115974 - cp's OEIS frontend

A000698 A problem of configurations: a(0) = 1; for n>0, a(n) = (2n-1)!! - Sum_{k=1..n-1} (2k-1)!! a(n-k). Also the number of shellings of an n-cube, divided by 2^n n!.

1, 1, 2, 10, 74, 706, 8162, 110410, 1708394, 29752066, 576037442, 12277827850, 285764591114, 7213364729026, 196316804255522, 5731249477826890, 178676789473121834, 5925085744543837186, 208256802758892355202, 7734158085942678174730

Offset: 0

N. J. A. Sloane, Richard Ehrenborg, Gabor Hetyei

Also number of nonisomorphic unlabeled connected Feynman diagrams of order 2n-2 for the electron propagator of quantum electrodynamics (QED), including vanishing diagrams. [Corrected by Charles R Greathouse IV, Jan 24 2014][Clarified by Robert Coquereaux, Sep 14 2014]
a(n+1) is the moment of order 2*n for the probability density function rho(x) = (1/sqrt(2*Pi))*exp(x^2/2)/[(u(x))^2+Pi/2], with u(x) = Integral_{t=0..x} exp(t*t/2) dt, on the real interval -infinity..infinity. - Groux Roland, Jan 13 2009
Starting (1, 2, 10, 74, ...) = INVERTi transform of A001147: (1, 3, 15, 105, ...). - Gary W. Adamson, Oct 21 2009
The Cvitanovic et al. paper relates this sequence to A005411 and A005413. - Robert Munafo, Jan 24 2010
Hankel transform of a(n+1) is A168467. - Paul Barry, Nov 26 2009
a(n) = number of labeled Dyck (n-1)-paths (A000108) in which each vertex that terminates an upstep is labeled with an integer i in [0,h], where h is the height of the vertex . For example UDUD contributes 4 labeled paths--0D0D, 0D1D, 1D0D, 1D1D where upsteps are replaced by their labels--and UUDD contributes 6 labeled paths to a(3)=10. The Deléham (Mar 24 2007) formula below counts these labeled paths by number of "0" labels. - David Callan, Aug 23 2011
a(n) is the number of indecomposable perfect matchings on [2n]. A perfect matching on [2n] is decomposable if a nonempty subset of the edges forms a perfect matching on [2k] for some kDavid Callan, Nov 29 2012
From Robert Coquereaux, Sep 12 2014: (Start)
QED diagrams are graphs with two kinds of edges (lines): a (non-oriented), f (oriented), and only one kind of (internal) vertex: aff. They may have internal and external (i.e., pendant) lines. The order is the number of (internal) vertices. Vanishing diagrams: QED diagrams containing loops of type f with an odd number of vertices are set to 0 (Furry theorem). Proper diagrams: connected QED diagrams that remain connected when an arbitrary internal line is cut.
The number of Feynman diagrams of order 2n for the electron propagator (2-point function of QED), vanishing or not, proper or not, of order 2n, starting from n = 0, is given by 1, 2, 10, 74, 706, 8162, ..., i.e., this sequence A000698, with the first term (equal to 1) dropped. Call Sf the associated g.f.
The number of non-vanishing Feynman diagrams, for the same 2-point function, is given by 1, 1, 4, 25, 208, 2146, ..., i.e., by the sequence A005411, with a first term of order 0, equal to 1, added. Call S the associated g.f.
If one does not remove the vanishing diagram, but, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams (vanishing and non-vanishing) for the self-energy function of QED, 0, 1, 3, 21, 207, 2529, ..., i.e., the sequence A115974 with a first term of order 0, equal to 0, added. A115974 is twice A167872. Call Sigmaf the associated g.f.
If one removes the vanishing diagrams and, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams for the self-energy function of QED given by 0, 1, 3, 18, 153, 1638, ..., i.e., by the sequence A005412, with a first term of order 0, equal to 0, added. Call Sigma the associated g.f.
Then Sf = 1/(1-Sigmaf) and S = 1/(1-Sigma). (End)
For n>0 sum over all Dyck paths of semilength n-1 of products over all peaks p of (x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 22 2015
Also, counts certain isomorphism classes of closed normal linear lambda terms. [N. Zeilberger, 2015]. - N. J. A. Sloane, Sep 18 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
For n >= 2, a(n) is the number of coalescent histories for a pair consisting of a matching lodgepole gene tree and species tree with 2n-1 leaves. - Noah A Rosenberg, Jun 21 2022

			G.f. = 1 + x + 2*x^2 + 10*x^3 + 74*x^4 + 706*x^5 + 8162*x^6 + 110410*x^7 + ...

Dubois C., Giorgetti A., Genestier R. (2016) Tests and Proofs for Enumerative Combinatorics. In: Aichernig B., Furia C. (eds) Tests and Proofs. TAP 2016. Lecture Notes in Computer Science, vol 9762. Springer.
R. W. Robinson, Counting irreducible Feynman diagrams exactly and asymptotically, Abstracts Amer. Math. Soc., 2002, #975-05-270.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces, Riccati's equation and continued fractions, Discrete Math., 215 (2000), 1-12.
F. Battaglia and T. F. George, A Pascal type triangle for the number of topologically distinct many-electron Feynman diagrams, J. Math. Chem. 2 (1988) 241-247.
S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, arXiv preprint arXiv:1111.4710 [math.CO], 2011.
S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, Discrete Math. 313 (2013), no. 3, 258-268. MR2995390.
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts)
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011, Lemma 3.2.
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Disc. Appl. Math. 161 (2013) 1378-1394
J. Courtiel, K. Yeats, and N. Zeilberger, Connected chord diagrams and bridgeless maps, arXiv:1661.04611, Proposition 13.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949, (eq 3.34 and fig 2b). DOI:10.1103/PhysRevD.18.1939
M. A. Deryagina and A. D. Mednykh, On the enumeration of circular maps with given number of edges, Siberian Mathematical Journal, 54, No. 6, 2013, 624-639.
F. Disanto and N. A. Rosenberg, Coalescent histories for lodgepole species trees, J. Comput. Biol. 22 (2015), 918-929.
S. Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv:1504.06904 [math.SP], 2015.
G. Edgar, Transseries for beginners, arXiv:0801.4877v5 [math.RA], 2008-2009.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Ali Assem Mahmoud, On the Asymptotics of Connected Chord Diagrams, University of Waterloo (Ontario, Canada 2019).
Ali Assem Mahmoud and Karen Yeats, Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions, arXiv:2010.06550 [math.CO], 2020.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.
R. J. Mathar, Table of Feynman Diagrams of the Interacting Fermion Green's Function, Int. J. Quant. Chem. 107 (2007) 1975. Also on arXiv, arXiv:physics/0512022 [physics.atom-ph], 2015-2016.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section VII.
L. G. Molinari, Hedin's equations and enumeration of Feynman diagrams, Phys. Rev. B, 71 (2005), 113102.
A. Prunotto, W. M. Alberico, and P. Czerski, Feynman diagrams and rooted maps, Open Phys. 16 (2018) 149-167.
J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.
J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25. [Annotated, corrected, scanned copy]
Wikipedia, Feynman diagram
Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, arXiv:math-ph/0303049, 2003.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A004208, A000165, A001147, A005412, row sums of A053979, A005411, A115974, A167872.
Column k=1 of A258219, A258222.
Row sums of A322398.

A006882 := proc(n) option remember; if n <= 1 then 1 else n*procname(n-2); fi; end;
A000698:=proc(n) option remember; global df; local k; if n=0 then RETURN(1); fi; A006882(2*n-1) - add(A006882(2*k-1)*A000698(n-k),k=1..n-1); end;
A000698 := proc(n::integer) local resul,fac,pows,c,c1,p,i ; if n = 0 then RETURN(1) ; else pows := combinat[partition](n) ; resul := 0 ; for p from 1 to nops(pows) do c := combinat[permute](op(p,pows)) ; c1 := op(1,c) ; fac := nops(c) ; for i from 1 to nops(c1) do fac := fac*doublefactorial(2*op(i,c1)-1) ; od ; resul := resul-(-1)^nops(c1)*fac ; od : fi ; RETURN(resul) ; end; # R. J. Mathar, Apr 24 2006
# alternative Maple program:
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> `if`(n=0, 1, b(2*n-2, 0, false)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 23 2015
a_list := proc(len) local n, A; if len=1 then return [1] fi: A := Array(-1..len-2); A[-1] := 1; A[0] := 1; for n to len-2 do A[n] := (2*n-1)*A[n-1]+add(A[j]*A[n-j-1], j=0..n-1) od: convert(A, list) end: a_list(20); # Peter Luschny, Jul 18 2017

a[n_] := a[n] = (2n - 1)!! - Sum[ a[n - k](2k - 1)!!, {k, n-1}]; Array[a, 18, 0] (* Ignacio D. Peixoto, Jun 23 2006 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 2 - 1 / Sum[ (2 k - 1)!! x^k, {k, 0, n}], {x, 0, n}]]; (* Michael Somos, Nov 16 2011 *)
a[n_]:= SeriesCoefficient[1+x(1/x+(E^((1/2)/x) Sqrt[2/\[Pi]] Sqrt[-(1/x)])/Erfc[Sqrt[-(1/x)]/Sqrt[2]]), {x,0,n}, Assumptions -> x >0](* Robert Coquereaux, Sep 14 2014 *)
max = 20; g = t/Fold[1 - ((t + #2)*z)/#1 &, 1, Range[max, 1, -1]]; T[n_, k_] := SeriesCoefficient[g, {z, 0, n}, {t, 0, k}]; a[0] = 1; a[n_] := Sum[T[n-1, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2016, after Philippe Deléham *)

{a(n) = if( n<0, 0, polcoeff( 2 - 1 / sum( k=0, n, x^k * (2*k)! /(2^k * k!), x * O(x^n)), n))}; /* Michael Somos, Feb 08 2011 */

{a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

from sympy import factorial2, cacheit
@cacheit
def a(n): return 1 if n == 0 else factorial2(2*n - 1) - sum(factorial2(2*k - 1)*a(n - k) for k in range(1, n))
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 18 2017

G.f.: 2 - 1/(1 + Sum_{n>=1} (2*n-1)!! * x^n ).
a(n+1) = Sum_{k=0..n} A089949(n, k)*2^k. - Philippe Deléham, Aug 15 2005
a(n+1) = Sum_{k=0..n} A053979(n,k). - Philippe Deléham, Mar 24 2007
From Paul Barry, Nov 26 2009: (Start)
G.f.: 1+x/(1-2x/(1-3x/(1-4x/(1-5x/(1-6x/(1-... (continued fraction).
G.f.: 1+x/(1-2x-6x^2/(1-7x-20x^2/(1-11x-42x^2/(1-15x-72x^2/(1-19x-110x^2/(1-... (continued fraction). (End)
G.f.: 1 + x * B(x) * C(x) where B(x) is the g.f. for A001147 and C(x) is the g.f. for A005416. - Michael Somos, Feb 08 2011
G.f.: 1+x/W(0); where W(k)=1+x+x*2k-x*(2k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
From Peter Bala, Dec 22 2011: (Start)
Recurrence relation: a(n+1) = (2*n-1)*a(n) + Sum_{k = 1..n} a(k)*a(n+1-k) for n >= 0 and a(1) = 1.
The o.g.f. B(x) = Sum_{n>=1} a(n)*x^(2*n-1) = x + 2*x^3 + 10*x^5 + 74*x^7 + ... satisfies the Riccati differential equation y'(x) = -1/x^2 + (1/x^3)*y(x) - (1/x^2)*y(x)^2 with initial condition y(0) = 0 (cf. A005412). The solution is B(x) = 1/z(x) + 1/x, where z(x) = -Sum_{n>=0} A001147(n) * x^(2*n+1) = -(x + x^3 + 3*x^5 + 15*x^7 + ...). The function b(x) = -B(1/x) satisfies b'(x) = -1 - (x + b(x))*b(x). Hence the differential operator (D^2 + x*D + 1), where D = d/dx, factorizes as (D - a(x))*(D - b(x)), where a(x) = -(x + b(x)), as conjectured by [Edgar, Problem 4.32]. For a refinement of this sequence see A053979. (End)
From Sergei N. Gladkovskii, Aug 19 2012, Oct 24 2012, Mar 19 2013, May 20 2013, May 29 2013, Aug 04 2013, Aug 05 2013: (Start)
Continued fractions:
G.f.: 2 - G(0) where G(k) = 1 - (k+1)*x/G(k+1).
G.f.: 2 - U(0) where U(k) = 1 - (2*k+1)*x/(1 - (2*k+2)*x/U(k+1)).
G.f.: 2 - U(0) where U(k) = 1 - (4*k+1)*x - (2*k+1)*(2*k+2)*x^2/U(k+1).
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+2)/(1 - x*(2*k+3)/Q(k+1)).
G.f.: 1 + x/Q(0) where Q(k) = 1 - x*(k+2)/Q(k+1).
G.f.: 2 - G(0)/2 where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/ G(k+1))).
G.f.: 1 + x*G(0) where G(k) = 1 - x*(k+2)/(x*(k+2) - 1/G(k+1)).
G.f.: 2 - 1/B(x) where B(x) is the g.f. of A001147.
G.f.: 1 + x/(1-2*x*B(x)) where B(x) is the g.f. of A167872. (End)
a(n) ~ 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 10 2014
G.f.: 1 + x*(1/x + (sqrt(2/Pi) * exp(1/(2*x)) * sqrt(-1/x))/Erfc(sqrt(-1/x)/sqrt(2))) where Erfc(z) = 1 - Erf(z) is the complementary error function, and Erf(z) is the integral of the Gaussian distribution. This generating function is obtained from the generating functional of (4-dimensional) QED, evaluated in dimension 0 for the 2-point function, without the modification implementing Furry theorem. - Robert Coquereaux, Sep 14 2014
From Peter Bala, May 23 2017: (Start)
G.f. A(x) = 1 + x/(1 + x - 3*x/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - ...)))).
A(x) = 1 + x/(1 + x - 3*x/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - ...))))))). (End)

Formula corrected by Ignacio D. Peixoto, Jun 23 2006

More terms from Sean A. Irvine, Feb 27 2011

A167872 A sequence of moments connected with Feynman numbers (A000698): Half the number of Feynman diagrams of order 2(n+1), for the electron self-energy in quantum electrodynamics (QED), i.e., all proper diagrams including Furry vanishing diagrams (those that vanish in 4-dimensional QED because of Furry theorem).

Original entry on oeis.org

1, 3, 21, 207, 2529, 36243, 591381, 10786527, 217179009, 4782674403, 114370025301, 2952426526767, 81864375589089, 2427523337157363, 76683680366193621, 2571609710380950207, 91265370849151405569, 3417956847888948899523

Offset: 0

Groux Roland, Nov 14 2009

nonn

a(n) is the moment of order 2*n of the probability density function defined by rho(x) = sqrt(Pi/2)*exp(-x^2/2)/((x*phi(x)+1)^2 + Pi^2*x^2*exp(-x^2)), where phi(x) = Integral_{t=-oo..oo} t*log(abs(x-t))*exp(-t^2/2) dt.

			G.f. = 1 + 3*x + 21*x^2 + 207*x^3 + 2529*x^4 + 36243*x^5 + 591381*x^6 + ...

Roland Groux. Polynômes orthogonaux et transformations intégrales. Cepadues. 2008. pages 195..206.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv preprint arXiv:1504.06904 [math.SP], 2015.
Adrian Ocneanu, On the inner structure of a permutation: bicolored partitions and Eulerians, trees and primitives; arXiv preprint arXiv:1304.1263 [math.CO], 2013.
Wikipedia, Feynman diagram

Row 3 of A321964. Cf. A000698, A115974, A005411, A005412, A001147.

(* f = A000698 *) f[n_] := f[n] = (2*n - 1)!! - Sum[f[n - k]*(2*k - 1)!!, {k, 1, n - 1}]; a[n_] := a[n] = f[n + 2]/2 - Sum[f[n + 1 - k]*a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 03 2013, from 3rd formula *)
nmax = 20; CoefficientList[Series[1/(1 + x + ContinuedFractionK[-(k - (-1)^k)*x, 1, {k, 3, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2022, after Peter Bala *)

{a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 23 2011 */

Sum_{n>=0} a(n)/z^(2n+1) = (1/2)*(z-S(z)/(z*S(z)-1)) with S(z) = Sum_{n>=0} (2*n)!/(2^n*n!*z^(2*n+1)).
a(n) = (2*n - 1) * a(n-1) + 2 * Sum_{k=1..n} a(k-1) * a(n-k) if n>0. - Michael Somos, Jul 23 2011
a(0)=1; for n > 0, a(n) = A000698(n+2)/2 - Sum_{k=0..n-1} A000698(n+1-k)*a(k).
G.f.: 1/(1-3*x/(1-4*x/(1-5*x/(1-6*x/(1-7*x/(1-8*x/(...))))))) (continued fraction). - Philippe Deléham, Nov 20 2011
G.f.: 1/Q(0), where Q(k) = 1 - x*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
Let A(x) be the g.f. of A127059 and B(x) be the g.f. of A167872. Then A(x) = (1 - 1/B(x))/x.
G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k+3)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - (2*k+3)*x/((2*k+2)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
G.f.: G(0), where G(k) = 1 - x*(k+3)/(x*(k+3) - 1/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Aug 05 2013
a(n) = A115974(n)/2, see comments in A115974. See also A000698, A005411, A005412. - Robert Coquereaux, Sep 14 2014
a(n) ~ 2^(n + 3/2) * n^(n+2) / exp(n). - Vaclav Kotesovec, Jan 02 2019
G.f.: 1/(1 + x - 4*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 8*x/(1 - 7*x/(1 - ...))))))). - Peter Bala, May 30 2022

Name clarified from Robert Coquereaux, Sep 14 2014

A127059 Column 2 of triangle A127058.

Original entry on oeis.org

3, 12, 108, 1332, 19908, 342252, 6583788, 139380372, 3211960068, 79950396492, 2137119431148, 61065403377012, 1858069709657028, 60006976422450732, 2050924514408985708, 73988085260209757652, 2810535115787602525188

Offset: 0

Paul D. Hanna, Jan 04 2007

nonn

Column 0 of triangle A127058 is A000698, the number of shellings of an n-cube, divided by 2^n n!. Column 1 of triangle A127058 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A127058; other columns: A000698, A115974; A127060.

A127058[n_, k_]:= A127058[n, k] = If[k==n, n+1, Sum[A127058[j+k, k]* A127058[n-j, k+1], {j,0,n - k - 1}]]; Table[A127058[n+2, 2], {n, 0, 30}] (* G. C. Greubel, Jun 09 2019 *)

c(n)=(2*n)!/(2^n*n!);
a(n)=if(n==0, 3, (c(n+3) - 3*c(n+2) - sum(k=0, n-1, a(k)*(c(n+2-k)-c(n+1-k)) ))/2  );
vector(20, n, n--; a(n)) \\ G. C. Greubel, Jun 09 2019

@CachedFunction
def A127058(n, k):
    if (k==n): return n+1
    else: return sum(A127058(j+k, k)*A127058(n-j, k+1) for j in (0..n-k-1))
[A127058(n+2,2) for n in (0..30)] # G. C. Greubel, Jun 09 2019

a(0) = 3 and for n>0 a(n) = (1/2)*(c(n+3)-3*c(n+2)-Sum_{k=0..n-1} a(k)*(c(n+2-k)-c(n+1-k))) with c(n) = (2*n)!/(2^n*n!). - Groux Roland, Nov 14 2009
G.f.: A(x) = (1 - T(0))/x, T(k) = 1 - x*(k+3)/T(k+1) (continued fraction). - Sergei N. Gladkovskii, Dec 13 2011
G.f.: 1/x - Q(0)/x, where Q(k)= 1 - x*(2*k+3)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
a(n) ~ 2^(n + 5/2) * n^(n+3) / exp(n). - Vaclav Kotesovec, Jan 02 2019

A127058 Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.

Original entry on oeis.org

1, 2, 2, 10, 6, 3, 74, 42, 12, 4, 706, 414, 108, 20, 5, 8162, 5058, 1332, 220, 30, 6, 110410, 72486, 19908, 3260, 390, 42, 7, 1708394, 1182762, 342252, 57700, 6750, 630, 56, 8, 29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9, 576037442

Offset: 0

Paul D. Hanna, Jan 04 2007

Column 0 is A000698, the number of shellings of an n-cube, divided by 2^n n!.

Column 1 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.

			Other recurrences exist, as shown by:
column 0 = A000698: T(n,0) = (2n+1)!! - Sum_{k=1..n} (2k-1)!!*T(n-k,0);
column 1 = A115974: T(n,1) = T(n+1,0) - Sum_{k=0..n-1} T(k,1)*T(n-k,0).
Illustrate the recurrence:
T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) (n > k >= 0)
at column k=1:
T(2,1) = T(1,1)*T(2,2) = 2*3 = 6;
T(3,1) = T(1,1)*T(3,2) + T(2,1)*T(2,2) = 2*12 + 6*3 = 42;
T(4,1) = T(1,1)*T(4,2) + T(2,1)*T(3,2) + T(3,1)*T(2,2) = 2*108 + 6*12 + 42*3 = 414;
at column k=2:
T(3,2) = T(2,2)*T(3,3) = 3*4 = 12;
T(4,2) = T(2,2)*T(4,3) + T(3,2)*T(3,3) = 3*20 + 12*4 = 108;
T(5,2) = T(2,2)*T(5,3) + T(3,2)*T(4,3) + T(4,2)*T(3,3) = 3*220 + 12*20 + 108*4 = 1332.
Triangle begins:
         1;
         2,        2;
        10,        6,       3;
        74,       42,      12,       4;
       706,      414,     108,      20,      5;
      8162,     5058,    1332,     220,     30,     6;
    110410,    72486,   19908,    3260,    390,    42,   7;
   1708394,  1182762,  342252,   57700,   6750,   630,  56,  8;
  29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9; ...

G. C. Greubel, Rows n = 0..15 of triangle, flattened

Columns: A000698, A115974, A127059.
Row sums: A127060.
Cf. A001147 ((2n-1)!!).

T[n_,k_]:= If[k==n, n+1, Sum[T[j+k,k]*T[n-j,k+1], {j,0,n-k-1}]];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)

{T(n,k)=if(n==k,n+1,sum(j=0,n-k-1,T(j+k,k)*T(n-j,k+1)))}

def T(n, k):
    if (k==n): return n+1
    else: return sum(T(j+k,k)*T(n-j,k+1) for j in (0..n-k-1))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 03 2019

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

cp's OEIS Frontend

A000698 A problem of configurations: a(0) = 1; for n>0, a(n) = (2n-1)!! - Sum_{k=1..n-1} (2k-1)!! a(n-k). Also the number of shellings of an n-cube, divided by 2^n n!.

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A127059 Column 2 of triangle A127058.

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A127058 Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.

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

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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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