cp's OEIS Frontend

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

%I A005413 M4445 #78 Mar 05 2023 03:12:26
%S A005413 1,1,7,72,891,12672,202770,3602880,70425747,1503484416,34845294582,
%T A005413 872193147840,23469399408510,676090493459712,20771911997290116,
%U A005413 678287622406488192,23466105907996232835,857623856612704266240
%N A005413 Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).
%C A005413 From _Robert Coquereaux_, Sep 12 2014: (Start)
%C A005413 QED diagrams are graphs with two kinds of edges (lines): a (non-oriented), f (oriented), and only one kind of (internal) vertex: aff.
%C A005413 They may have internal and external (i.e., pendant) lines.
%C A005413 QED diagrams containing loops of type f with an odd number of vertices are set to 0 (vanishing diagrams).
%C A005413 Proper diagrams also called one-particle-irreducible diagrams (1PI) are connected diagrams that remain connected when an arbitrary internal line is cut.
%C A005413 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.
%C A005413 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.
%C A005413 Write g*Gamma(g) = g (1 + x + 7 x2 + 72 x3 + ...) with x = g^2.
%C A005413 The sequence a(n) gives the coefficient of x^n.
%C A005413 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.
%C A005413 (End)
%D A005413 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005413 C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.
%H A005413 Robert Coquereaux, <a href="/A005413/b005413.txt">Table of n, a(n) for n = 0..250</a>
%H A005413 P. Cvitanovic, B. Lautrup and R. B. Pearson, <a href="https://cns.gatech.edu/~predrag/papers/PRD18-78.pdf">The number and weights of Feynman diagrams</a>, Phys. Rev. D18, pp. 1939-1949 (1978).
%H A005413 Kevin Hartnett, <a href="https://www.quantamagazine.org/20161115-strange-numbers-found-in-particle-collisions/">Physicists uncover strange numbers in particle collisions</a>, Quanta Magazine, November 15 2016.
%H A005413 R. J. Martin and M. J. Kearney, <a href="http://dx.doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318. see p. 310.
%H A005413 R. J. Martin and M. J. Kearney, <a href="http://arXiv.org/abs/1103.4936">An exactly solvable self-convolutive recurrence</a>, arXiv:1103.4936 [math.CO], 2011.
%H A005413 Wikipedia, <a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</a>
%F A005413 See recurrence in Martin-Kearney paper.
%F A005413 a(n) = (n-1)*(A005412(n) + 2*n*A005412(n-1)) if n > 1.
%F A005413 From _Robert Coquereaux_, Sep 12 2014: (Start)
%F A005413 The g.f. for this sequence is (U - 1)/(U^3 x) where U is the g.f. for A005411.
%F A005413 G.f.: (4*x*(-2*x + (1 - K(1, -(1/(4*x))) / K(0, -(1/(4*x))))))/
%F A005413    (1 - K(1, -(1/(4*x))) / K(0, -(1/(4*x))))^3
%F A005413 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".
%F A005413 (End)
%e A005413 G.f. = 1 + x + 7*x^2 + 72*x^3 + 891*x^4 + 12672*x^5 + 202770*x^6 + 3602880*x^7 + ...
%t A005413 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 *)
%o A005413 (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 */
%o A005413 (Haskell)
%o A005413 a005413 n = a005413_list !! (n-1)
%o A005413 a005413_list = 1 : zipWith (*) [1 ..]
%o A005413                            (zipWith (+) (tail a005412_list)
%o A005413                            (zipWith (*) [4, 6 ..] a005413_list))
%o A005413 -- _Reinhard Zumkeller_, Jan 24 2014
%Y A005413 Cvitanovic et al. relate this sequence to A000698 and A005411. - _Robert Munafo_, Jan 24 2010
%K A005413 nonn,easy
%O A005413 0,3
%A A005413 _N. J. A. Sloane_
%E A005413 Name clarified and reference added by _Robert Coquereaux_, Sep 12 2014