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A005411 Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.
(Formerly M3610)
7
1, 4, 25, 208, 2146, 26368, 375733, 6092032, 110769550, 2232792064, 49426061818, 1192151302144, 31123028996164, 874428204384256, 26308967412122125, 843984969276915712, 28757604639850111894, 1037239628039528906752, 39481325230750749160462 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 counts 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)

REFERENCES

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).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..200

P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D 18, 1939-1949 (1978).

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO]

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 294.

FORMULA

From Peter Bala, Mar 07 2011: (Start)

Given o.g.f. A(x), then function F(x) := 1 + 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:

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 + 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

EXAMPLE

G.f. = x + 4*x^2 + 25*x^3 + 208*x^4 + 2146*x^5 + 26368*x^6 + 375733*x^7 + ...

MATHEMATICA

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-Fran├žois Alcover, Feb 27 2014, after Michael Somos *)

a[ n_] := Module[{m = n + 1, u}, If[ n < 2, Boole[n == 1], 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 *)

PROG

(PARI) {a(n) = local(A); if( n<1, 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 */

CROSSREFS

Cf. A094664.

Sequence in context: A036242 A120955 A061714 * A105628 A203219 A064299

Adjacent sequences:  A005408 A005409 A005410 * A005412 A005413 A005414

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Name corrected by Charles R Greathouse IV, Jan 24 2014

Name clarified by Robert Coquereaux, Sep 05 2014

STATUS

approved

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Last modified November 26 17:56 EST 2014. Contains 250082 sequences.