A005412 Number of non-vanishing Feynman diagrams of order 2n for the vacuum polarization (the proper two-point function of the photon) and for the self-energy (the proper two-point function of the electron) in quantum electrodynamics (QED).

1, 3, 18, 153, 1638, 20898, 307908, 5134293, 95518278, 1961333838, 44069970348, 1075902476058, 28367410077468, 803551902237828, 24342558819042888, 785445178323709773, 26896354975287884358, 974297972094661642518, 37225733779871789177628, 1496237868417003741147438

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

There was a typo in the value of a(10) = 1967333838 previously given in the database (taken from the self-energies column of Table 1 in P. Cvitanovic et al.). The corrected value is given above. - Peter Bala, Mar 07 2011

From Robert Coquereaux, Sep 12 2014: (Start)

Proper diagrams also called one-particle-irreducible diagrams (1PI) are connected diagrams that remain connected when an arbitrary internal line is cut (see the Comments in A005413 for other terminological details).

The number of non-vanishing Feynman diagrams for these two functions (see Name field) is the same. It is given by the coefficients of Sigma(g) = g^2 + 3*g^4 + 18*g^6 + 153*g^8 + ... where the exponent p of g^p refers to the number of (internal) vertices. Setting x=g^2, the sequence a(n) gives the coefficient of x^n.

If one relaxes the "proper" condition, the number of non-vanishing Feynman diagrams for the corresponding (complete) two-point functions, also called propagators, is given by 1,1,4,25,208,..., i.e., by the sequence A005411 with offset 0 and A005411(0)=1. The relation between the two is given by Sigma(g) = 1 - 1/S(g) where S(g) is defined by A005411 as S(g) = 1 + g^2 + 4*g^4 + 25*g^6 + ...

(End)

For n > 0 sum over all Dyck paths of semilength n-1 of products over all peaks p of (x_p+2*y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 22 2015

Examples

			x + 3*x^2 + 18*x^3 + 153*x^4 + 1638*x^5 + 20898*x^6 + 307908*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

• Reinhard Zumkeller, Table of n, a(n) for n = 1..250
• 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. 293.
• A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
• Wikipedia, Feynman diagram

Crossrefs

Cf. A005411.

Column k=2 of A258219.

Programs

Haskell

a005412 n = a005412_list !! (n-1)
a005412_list = 1 : f 2 [1] where
   f v ws@(w:_) = y : f (v + 2) (y : ws) where
                  y = v * w + (sum $ zipWith (*) ws $ reverse ws)
-- Reinhard Zumkeller, Jan 24 2014

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+2*y)/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n-2, 0, false):
seq(a(n), n=1..25);  # Alois P. Heinz, May 23 2015

Mathematica

a[n_]:=SeriesCoefficient[1 - (2*x)/(1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))]),{x,0,n}] (* Robert Coquereaux, Sep 12 2014 *)
Clear[a]; a[1] = 1; a[n_]:= a[n] = (2*n-2)*a[n-1] + Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

PARI

{a(n) = local(A); if( n<1, 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])); A[n])} /* Michael Somos, Jul 23 2011 */

Formula

See recurrence in Martin-Kearney paper.

From Peter Bala, Mar 07 2011: (Start)

The o.g.f. A(x) = x^2 + 3*x^4 + 18*x^6 + 153*x^8 + ... satisfies the differential equation A(x) = x^2 + x^3*A'(x) + A(x)^2 (equation 3.55, P. Cvitanovic et al., A'(x) the derivative of A(x)).

Conjectural o.g.f. as a continued fraction:

x^2/(1-3*x^2/(1-3*x^2/(1-5*x^2/(1-5*x^2/(1-7*x^2/(1-7*x^2/(1-...))))))).

[follows by applying the result of Stokes to the g.f. G(x) := (1/x)*A(sqrt(x)), which satisfies the Riccati differential equation 2*x^2*G'(x) + 1 + (2*x - 1)*G(x) + x*G^2(x) = 0 - added by Peter Bala, Jun 22 2022]. (End).

a(n) = (2*n - 2) * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011

G.f.: 1/x - Q(0)/x, where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+3)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

G.f.: 1/x - 2 - Q(0)/x, where Q(k) = 1 - x*(2*k+3)/(1 - x*(2*k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013

G.f.: 1/x + 1/( Q(0)-1 ), where Q(k) = 1 - (2*k+1)*x/(1 - (2*k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 18 2013

G.f.: 1/x - Q(0)/x, where Q(k) = 1 + x*(2*k+2) - (2*k+3)*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013

From the relation with A005411, one finds the g.f.: 1 - (2*x)/(1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))]). - Robert Coquereaux, Sep 12 2014

This satisfies the d.e. 2*x^2*g'(x) - g(x) + g(x)^2 = -x, which can be obtained from the d.e. for A(x) by A(sqrt(x)) = g(x). - Robert Israel, Sep 12 2014

a(n) ~ 2^(n+1) * n! / Pi. - Vaclav Kotesovec, Jan 19 2015

Extensions

Name corrected by Charles R Greathouse IV, Jan 24 2014

Name clarified by Robert Coquereaux, Sep 12 2014