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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).
(Formerly M3050)
21
1, 3, 18, 153, 1638, 20898, 307908, 5134293, 95518278, 1961333838, 44069970348, 1075902476058, 28367410077468, 803551902237828, 24342558819042888, 785445178323709773, 26896354975287884358, 974297972094661642518 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

REFERENCES

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

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, 1939 (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. 293.

Wikipedia, Feynman diagram

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-...))))))).

(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

EXAMPLE

x + 3*x^2 + 18*x^3 + 153*x^4 + 1638*x^5 + 20898*x^6 + 307908*x^7 + ...

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 *)

PROG

(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 */

(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

CROSSREFS

Cf. A005411.

Column k=2 of A258219.

Sequence in context: A152409 A200320 A207569 * A145350 A107888 A293491

Adjacent sequences:  A005409 A005410 A005411 * A005413 A005414 A005415

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 12 2014

STATUS

approved

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Last modified February 21 04:22 EST 2018. Contains 299389 sequences. (Running on oeis4.)