

A005412


Number of nonvanishing Feynman diagrams of order 2n for the vacuum polarization (the proper twopoint function of the photon) and for the selfenergy (the proper twopoint 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
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OFFSET

1,2


COMMENTS

There was a typo in the value of a(10) = 1967333838 previously given in the database (taken from the selfenergies 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 oneparticleirreducible 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 nonvanishing 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 nonvanishing Feynman diagrams for the corresponding (complete) twopoint 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) = 11/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 n1 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

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, McGrawHill, 1980, pages 466467.


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), 19391949, [DOI]
R. J. Martin and M. J. Kearney, An exactly solvable selfconvolutive recurrence, arXiv:1103.4936 [math.CO]
R. J. Martin and M. J. Kearney, An exactly solvable selfconvolutive recurrence, Aequat. Math., 80 (2010), 291318. see p. 293.
Wikipedia, Feynman diagram


FORMULA

See recurrence in MartinKearney 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/(13*x^2/(13*x^2/(15*x^2/(15*x^2/(17*x^2/(17*x^2/(1...))))))).
(End).
a(n) = (2*n  2) * a(n1) + Sum_{k=1..n1} a(k) * a(nk) 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(x1, y1, false)*`if`(t, (x+2*y)/y, 1) +
b(x1, y+1, true) ))
end:
a:= n> b(2*n2, 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*n2)*a[n1] + Sum[a[k]*a[nk], {k, 1, n1}]; 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[k1] + sum( j=1, k1, A[j] * A[kj])); A[n])} /* Michael Somos, Jul 23 2011 */
(Haskell)
a005412 n = a005412_list !! (n1)
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



