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Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.
(Formerly M3610)
25

%I M3610 #170 Aug 30 2023 09:19:42

%S 1,1,4,25,208,2146,26368,375733,6092032,110769550,2232792064,

%T 49426061818,1192151302144,31123028996164,874428204384256,

%U 26308967412122125,843984969276915712,28757604639850111894,1037239628039528906752,39481325230750749160462

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

%C Cvitanovic et al. paper relates this sequence to A000698 and A005413. - _Robert Munafo_, Jan 24 2010

%C (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.

%C 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

%C From _Robert Coquereaux_, Sep 05 2014: (Start)

%C 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.

%C These are graphs with two kinds of edges: a (non-oriented), f (oriented), and only one kind of vertex: aff.

%C Electron propagator: all the diagrams with two external edges of type f.

%C Photon propagator: all the diagrams with two external edges of type a.

%C The exponent n of g^n gives the number of vertices.

%C Diagrams containing loops of type f with an odd number of vertices are set to 0 (vanishing diagrams).

%C 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.

%C 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).

%C (End)

%C 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

%C From _Sasha Kolpakov_, Dec 11 2017: (Start)

%C Number of free index 2n subgroups in the free product Z_2*Z_2*Z_2.

%C Number of oriented rooted pavings (after Arques & Koch, Spehner, Lienhardt) with 2n darts.

%C (End)

%D C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A005411/b005411.txt">Table of n, a(n) for n = 0..400</a>

%H Michael Borinsky, <a href="https://arxiv.org/abs/1703.00840">Renormalized asymptotic enumeration of Feynman diagrams</a>, arXiv:1703.00840 [hep-th], 2017.

%H Rémi Bottinelli, Laura Ciobanu, and Alexander Kolpakov, <a href="https://doi.org/10.1007/s00229-021-01321-7">Three-dimensional maps and subgroup growth</a>, manuscripta math. (2021).

%H L. Ciobanu and A. Kolpakov, <a href="https://arxiv.org/abs/1712.01418">Three-dimensional maps and subgroup growth</a>, arXiv:1712.01418 [math.GR], 2017.

%H P. Cvitanovic, B. Lautrup and R. B. Pearson, <a href="https://cns.gatech.edu/~predrag/papers/PRD18-78.pdf">The number and weights of Feynman diagrams</a>, Phys. Rev. D18, (1978), 1939-1949. <a href="https://doi.org/10.1103/PhysRevD.18.1939">DOI:10.1103/PhysRevD.18.1939</a>

%H R. J. Martin and M. J. Kearney, <a href="http://arXiv.org/abs/1103.4936">An exactly solvable self-convolutive recurrence</a>, arXiv:1103.4936 [math.CO], 2011.

%H R. J. Martin and M. J. Kearney, <a href="http://dx.doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318. see p. 294.

%H A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</a>

%F From _Peter Bala_, Mar 07 2011: (Start)

%F 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.).

%F Conjectural o.g.f. A(x) as a continued fraction:

%F 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 - ...)))).

%F Asymptotics: a(n) ~ 1/Pi*2^(n+1)*n!*(1 - 1/(2*n) - 3/(8*n^2)). (End)

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F From _Peter Bala_, Jun 27 2022: (Start)

%F 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).

%F 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)

%e 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]

%p b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,

%p `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x/y, 1) +

%p b(x-1, y+1, true) ))

%p end:

%p a:= n-> b(2*n, 0, false):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 21 2015

%t 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_ *)

%t 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 *)

%t 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 *)

%o (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 */

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

%Y Column k=0 of A258219, A258220.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E Name corrected by _Charles R Greathouse IV_, Jan 24 2014

%E Name clarified by _Robert Coquereaux_, Sep 05 2014

%E a(0)=1 prepended, programs and formulas edited by _Alois P. Heinz_, Jun 22 2015