A005413 Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).

1, 1, 7, 72, 891, 12672, 202770, 3602880, 70425747, 1503484416, 34845294582, 872193147840, 23469399408510, 676090493459712, 20771911997290116, 678287622406488192, 23466105907996232835, 857623856612704266240

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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.

QED diagrams containing loops of type f with an odd number of vertices are set to 0 (vanishing diagrams).

Proper diagrams also called one-particle-irreducible diagrams (1PI) are connected diagrams that remain connected when an arbitrary internal line is cut.

The proper vertex function of QED is described by proper (1PI) diagrams with one external line of type a (photon) and two external lines of type f (electron). Non-vanishing diagrams only exist if the number of vertices is odd.

The number of non-vanishing Feynman diagrams for the proper vertex function is obtained from g*Gamma(g) = g (1 + 1 g^2 + 7 g^4 + 72 g^6 + ...) where the exponent p of g^p gives the number of (internal) vertices, p is called the order of the diagram.

Write g*Gamma(g) = g (1 + x + 7 x2 + 72 x3 + ...) with x = g^2.

The sequence a(n) gives the coefficient of x^n.

Relation with A005411: Gamma (g) = (S(g) - 1)/(g^2 S(g)^3) where S(g) = 1 + g^2 + 4 g^4 + 25 g^6 + ... is sum A005411(n) g^(2n), hence the g.f. in terms of modified Bessel functions.

(End)

Examples

			G.f. = 1 + x + 7*x^2 + 72*x^3 + 891*x^4 + 12672*x^5 + 202770*x^6 + 3602880*x^7 + ...
		

References

• 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.

Links

• Robert Coquereaux, Table of n, a(n) for n = 0..250
• P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, pp. 1939-1949 (1978).
• Kevin Hartnett, Physicists uncover strange numbers in particle collisions, Quanta Magazine, November 15 2016.
• R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 310.
• R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
• Wikipedia, Feynman diagram

Crossrefs

Cvitanovic et al. relate this sequence to A000698 and A005411. - Robert Munafo, Jan 24 2010

Programs

Haskell

a005413 n = a005413_list !! (n-1)
a005413_list = 1 : zipWith (*) [1 ..]
                           (zipWith (+) (tail a005412_list)
                           (zipWith (*) [4, 6 ..] a005413_list))
-- Reinhard Zumkeller, Jan 24 2014

Mathematica

a[n_]:= SeriesCoefficient[(4*x*(-2*x + (1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))])))/ (1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))])^3, {x,0,n}] (* Robert Coquereaux, Sep 12 2014 *)

PARI

{a(n) = my(A); if( n<2, n>=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])); (n-1) * (A[n] + 2 * n * A[n-1]))}; /* Michael Somos, Jul 24 2011 */

Formula

See recurrence in Martin-Kearney paper.

a(n) = (n-1)*(A005412(n) + 2*n*A005412(n-1)) if n > 1.

From Robert Coquereaux, Sep 12 2014: (Start)

The g.f. for this sequence is (U - 1)/(U^3 x) where U is the g.f. for A005411.

G.f.: (4*x*(-2*x + (1 - K(1, -(1/(4*x))) / K(0, -(1/(4*x))))))/

(1 - K(1, -(1/(4*x))) / K(0, -(1/(4*x))))^3

where K(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 the 1980 book "Quantum Field Theory".

(End)

Extensions

Name clarified and reference added by Robert Coquereaux, Sep 12 2014