

A005413


Number of nonvanishing Feynman diagrams of order 2n+1 for the electronelectronphoton proper vertex function in quantum electrodynamics (QED).
(Formerly M4445)


6



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

0,3


COMMENTS

From Robert Coquereaux, Sep 12 2014: (Start)
QED diagrams are graphs with two kinds of edges (lines): a (nonoriented), f (oriented), and only one kind of (internal) vertex: aff.
They may have internal and external (ie 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 oneparticleirreducible 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). Nonvanishing diagrams only exist if the number of vertices is odd.
The number of nonvanishing 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)


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, McGrawHill, 1980, pages 466467.


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. 19391949 (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 selfconvolutive recurrence, Aequat. Math., 80 (2010), 291318. see p. 310.
R. J. Martin and M. J. Kearney, An exactly solvable selfconvolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
Wikipedia, Feynman diagram


FORMULA

See recurrence in MartinKearney 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 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)


EXAMPLE

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


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 *)


PROG

(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[k1] + sum( j=1, k1, A[j] * A[kj])); (n1) * (A[n] + 2 * n * A[n1]))}; /* Michael Somos, Jul 24 2011 */
(Haskell)
a005413 n = a005413_list !! (n1)
a005413_list = 1 : zipWith (*) [1 ..]
(zipWith (+) (tail a005412_list)
(zipWith (*) [4, 6 ..] a005413_list))
 Reinhard Zumkeller, Jan 24 2014


CROSSREFS

Cvitanovic et al. relate this sequence to A000698 and A005411.  Robert Munafo, Jan 24 2010
Sequence in context: A242827 A077332 A222833 * A306377 A230890 A203473
Adjacent sequences: A005410 A005411 A005412 * A005414 A005415 A005416


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

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


STATUS

approved



