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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 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)

Examples

			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]
		

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 = 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

Crossrefs

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

Column k=0 of A258219, A258220.

Programs

Maple

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

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-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 *)

PARI

{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 */

Formula

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)

Extensions

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