A258219 -id:A258219 - 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

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

Original entry on oeis.org

1, 1, 4, 25, 208, 2146, 26368, 375733, 6092032, 110769550, 2232792064, 49426061818, 1192151302144, 31123028996164, 874428204384256, 26308967412122125, 843984969276915712, 28757604639850111894, 1037239628039528906752, 39481325230750749160462

Offset: 0

N. J. A. Sloane

Cvitanovic et al. paper relates this sequence to A000698 and A005413. - Robert Munafo, Jan 24 2010
(x + 4x^2 + 25x^3 + 208x^4 + ...) = (x + 2x^2 + 7x^3 + 38x^4 + ...) * 1/(1 + x + 2x^2 + 7x^3 + 38x^4 + ...); where A094664 = (1, 1, 2, 7, 38, 286, ...). - Gary W. Adamson, Nov 16 2011.
The Martin and Kearney article has S(2,-4,1) = [1,1,4,25,...] where u_1 = u_2 = 1, u_3 = 4, u_4 =25, etc. This is almost the same as this sequence. - Michael Somos, Feb 27 2014
From Robert Coquereaux, Sep 05 2014: (Start)
Evaluation of quantum electrodynamics functional integrals in dimension 0 become usual Lebesgue integrals, their Taylor expansion around g=0 at order n give the number of Feynman diagrams.
These are graphs with two kinds of edges: a (non-oriented), f (oriented), and only one kind of vertex: aff.
Electron propagator: all the diagrams with two external edges of type f.
Photon propagator: all the diagrams with two external edges of type a.
The exponent n of g^n gives the number of vertices.
Diagrams containing loops of type f with an odd number of vertices are set to 0 (vanishing diagrams).
The coefficients of the series S(g)=Sum a(n) g^(2n) give the number of non-vanishing Feynman diagrams for the electron (or the photon) propagator.
S(g) is obtained as < 1/(1-g^2 a^2) > for the measure (E^(-(a^2/2)))/sqrt[1-g^2 a^2]da, assuming g^2 < 0, hence a formula for S(g) in terms of modified Bessel functions (setting x=g^2 gives the G.f. below).
(End)
Sum over all Dyck paths of semilength n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p. a(3) = 3/3 +2/2*5/1 +1/1*4/2 +2/2*4/2 +1/1*3/1*5/1 = 25. - Alois P. Heinz, May 21 2015
From Sasha Kolpakov, Dec 11 2017: (Start)
Number of free index 2n subgroups in the free product Z_2*Z_2*Z_2.
Number of oriented rooted pavings (after Arques & Koch, Spehner, Lienhardt) with 2n darts.
(End)

			G.f. = 1 + x + 4*x^2 + 25*x^3 + 208*x^4 + 2146*x^5 + 26368*x^6 + 375733*x^7 + ... [Deleted g.f. restored by _N. J. A. Sloane_, Jan 30 2016]

C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..400
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
Rémi Bottinelli, Laura Ciobanu, and Alexander Kolpakov, Three-dimensional maps and subgroup growth, manuscripta math. (2021).
L. Ciobanu and A. Kolpakov, Three-dimensional maps and subgroup growth, arXiv:1712.01418 [math.GR], 2017.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949. DOI:10.1103/PhysRevD.18.1939
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. 294.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Wikipedia, Feynman diagram

Cf. A000108, A000698, A005412, A005413, A094664.

Column k=0 of A258219, A258220.

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, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);  # Alois P. Heinz, May 21 2015

a[n_] := Module[{A}, A[1] = 1; A[k_] := A[k] = (2*k-4)*A[k-1]+Sum[A[j]*A[k-j], {j, 1, k-1}]; A[n]]; Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Feb 27 2014, after Michael Somos *)
a[ n_] := Module[{m = n + 1, u}, If[ n < 2, Boole[n >= 0], u = Range[m]; Do[ u[[k]] = (2 k - 4) u[[k - 1]] + Sum[ u[[j]] u[[k - j]], {j, k - 1}], {k, 2, m}]; u[[m]]]]; (* Michael Somos, Feb 27 2014 *)
a[n_]:=SeriesCoefficient[(1-BesselK[1,-(1/(4 g^2))]/BesselK[0,-(1/(4 g^2))])/(2 g^2),{g,0,2*n}]; (* Robert Coquereaux, Sep 05 2014 *)

{a(n) = my(A); if( n<1, n==0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2 * k - 4) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

From Peter Bala, Mar 07 2011: (Start)
Given the o.g.f. A(x), the function F(x) := A(x^2) satisfies the differential equation F(x) = 1 + x^3*d/dx(F(x)) + x^2*F(x)^2 (equation 3.53, P. Cvitanovic et al.).
Conjectural o.g.f. A(x) as a continued fraction:
1 + x/(1 - 4*x - 3^2*x^2/(1 - 8*x - 5^2*x^2/(1 - 12*x - 7^2*x^2/(1 - 16*x - ...)))).
Asymptotics: a(n) ~ 1/Pi*2^(n+1)*n!*(1 - 1/(2*n) - 3/(8*n^2)). (End)
Given u(1) = 1, u(n) = (2*n - 4) * u(n-1) + Sum_{k=1..n-1} u(k) * u(n-k) when n>1, then a(n) = u(n+1) if n>0. - Michael Somos, Jul 24 2011
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1/x^2 - 1/x - Q(0)/x^2, where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
G.f.: 1/x^2 - 1/x - G(0)/(2*x^2), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(2*k+1)/( x*(2*k+1) - 1/(1 - x*(2*k+3)/( x*(2*k+3) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
G.f.: G(0)/x -1/x, where G(k) = 1 - x*(2*k+1)/(x - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2014
G.f.: 1/(2*x) - BesselK(1,-1/(4*x))/(2*x*BesselK(0,-1/(4*x))) where BesselK[p,z] denotes the modified Bessel function of the second kind (order p, argument z). This is a small improvement of a result obtained in 1980 book "Quantum Field Theory". - Robert Coquereaux, Sep 05 2014
Asymptotics: a(n) ~ 2*(2/Pi)^(1/2)*(2/e)^n*n^(n+1/2), cf. Ciobanu and Kolpakov in Links. - Sasha Kolpakov, Dec 11 2017
From Peter Bala, Jun 27 2022: (Start)
O.g.f. as a continued fraction of Stieltjes type: A(x) = 1/(1 - x/(1 - 3*x/(1 - 3*x/(1 - 5*x/(1 - 5*x/(1 - 7*x/(1 - 7*x/(1 - ...)))))))) follows by applying the result of Stokes to the Riccati differential equation 2*x^2*A'(x) = -1 + A(x) - x*A^2(x).
The even part of the continued fraction gives A(x) = 1/(1 - x - 3*x^2/(1 - 6*x - 15*x^2/(1 - 10*x - 35*x^2/(1 - 14*x - 63*x^2/(1 - 18*x - ... - (4*n^2-1)*x^2/(1 - (4*n+2)*x -...)))))), a continued fraction of Jacobi type (a J-fraction). (End)

Name corrected by Charles R Greathouse IV, Jan 24 2014
Name clarified by Robert Coquereaux, Sep 05 2014
a(0)=1 prepended, programs and formulas edited by Alois P. Heinz, Jun 22 2015

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).

Original entry on oeis.org

1, 3, 18, 153, 1638, 20898, 307908, 5134293, 95518278, 1961333838, 44069970348, 1075902476058, 28367410077468, 803551902237828, 24342558819042888, 785445178323709773, 26896354975287884358, 974297972094661642518, 37225733779871789177628, 1496237868417003741147438

Offset: 1

N. J. A. Sloane

There was a typo in the value of a(10) = 1967333838 previously given in the database (taken from the self-energies column of Table 1 in P. Cvitanovic et al.). The corrected value is given above. - Peter Bala, Mar 07 2011
From Robert Coquereaux, Sep 12 2014: (Start)
Proper diagrams also called one-particle-irreducible diagrams (1PI) are connected diagrams that remain connected when an arbitrary internal line is cut (see the Comments in A005413 for other terminological details).
The number of non-vanishing Feynman diagrams for these two functions (see Name field) is the same. It is given by the coefficients of Sigma(g) = g^2 + 3*g^4 + 18*g^6 + 153*g^8 + ... where the exponent p of g^p refers to the number of (internal) vertices. Setting x=g^2, the sequence a(n) gives the coefficient of x^n.
If one relaxes the "proper" condition, the number of non-vanishing Feynman diagrams for the corresponding (complete) two-point functions, also called propagators, is given by 1,1,4,25,208,..., i.e., by the sequence A005411 with offset 0 and A005411(0)=1. The relation between the two is given by Sigma(g) = 1 - 1/S(g) where S(g) is defined by A005411 as S(g) = 1 + g^2 + 4*g^4 + 25*g^6 + ...
(End)
For n > 0 sum over all Dyck paths of semilength n-1 of products over all peaks p of (x_p+2*y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 22 2015

			x + 3*x^2 + 18*x^3 + 153*x^4 + 1638*x^5 + 20898*x^6 + 307908*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949. DOI:10.1103/PhysRevD.18.1939
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. 293.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Wikipedia, Feynman diagram

Cf. A005411.

Column k=2 of A258219.

a005412 n = a005412_list !! (n-1)
a005412_list = 1 : f 2 [1] where
   f v ws@(w:_) = y : f (v + 2) (y : ws) where
                  y = v * w + (sum $ zipWith (*) ws $ reverse ws)
-- Reinhard Zumkeller, Jan 24 2014

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+2*y)/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n-2, 0, false):
seq(a(n), n=1..25);  # Alois P. Heinz, May 23 2015

a[n_]:=SeriesCoefficient[1 - (2*x)/(1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))]),{x,0,n}] (* Robert Coquereaux, Sep 12 2014 *)
Clear[a]; a[1] = 1; a[n_]:= a[n] = (2*n-2)*a[n-1] + Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

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

See recurrence in Martin-Kearney paper.
From Peter Bala, Mar 07 2011: (Start)
The o.g.f. A(x) = x^2 + 3*x^4 + 18*x^6 + 153*x^8 + ... satisfies the differential equation A(x) = x^2 + x^3*A'(x) + A(x)^2 (equation 3.55, P. Cvitanovic et al., A'(x) the derivative of A(x)).
Conjectural o.g.f. as a continued fraction:
x^2/(1-3*x^2/(1-3*x^2/(1-5*x^2/(1-5*x^2/(1-7*x^2/(1-7*x^2/(1-...))))))).
[follows by applying the result of Stokes to the g.f. G(x) := (1/x)*A(sqrt(x)), which satisfies the Riccati differential equation 2*x^2*G'(x) + 1 + (2*x - 1)*G(x) + x*G^2(x) = 0 - added by Peter Bala, Jun 22 2022]. (End).
a(n) = (2*n - 2) * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
G.f.: 1/x - Q(0)/x, where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+3)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
G.f.: 1/x - 2 - Q(0)/x, where Q(k) = 1 - x*(2*k+3)/(1 - x*(2*k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
G.f.: 1/x + 1/( Q(0)-1 ), where Q(k) = 1 - (2*k+1)*x/(1 - (2*k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 18 2013
G.f.: 1/x - Q(0)/x, where Q(k) = 1 + x*(2*k+2) - (2*k+3)*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013
From the relation with A005411, one finds the g.f.: 1 - (2*x)/(1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))]). - Robert Coquereaux, Sep 12 2014
This satisfies the d.e. 2*x^2*g'(x) - g(x) + g(x)^2 = -x, which can be obtained from the d.e. for A(x) by A(sqrt(x)) = g(x). - Robert Israel, Sep 12 2014
a(n) ~ 2^(n+1) * n! / Pi. - Vaclav Kotesovec, Jan 19 2015

Name corrected by Charles R Greathouse IV, Jan 24 2014

Name clarified by Robert Coquereaux, Sep 12 2014

cp's OEIS Frontend

A258220 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258219(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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

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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).

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A258222 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A292693 Sum over all Dyck paths of semilength n of products over all peaks p of (x_p+n*y_p)/y_p, where x_p and y_p are the coordinates of peak p.

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A258220 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) C(k,i) A258219(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.