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A005414 Feynman diagrams of order 2n with vertex skeletons.
(Formerly M4906)
2
1, 1, 13, 93, 1245, 18093, 308605, 5887453, 124221373, 2864305277, 71589605885, 1927010749181, 55572839581437, 1709604517055229, 55893262628149245, 1935654236127347709, 70799043456576835581, 2727771901780930132989, 110438840436968476274685, 4688223534904569925386237 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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).

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..202

Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.

P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, 1939 (1978).

FORMULA

a(n) ~ 4*exp(-5/2)/Pi * n * 2^n * n! * (1 - 9/(4*n) - 67/(32*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 19 2017

MATHEMATICA

seq[nn_] := Module[{x, y=0, y1=0, n=1}, While[n++; True, y1 = x^2 + x^4 + 2x^6 - 3x^2 y + x^4 (-y + x D[y, x]/2) - x^6 (8y + x D[y, x]/2) + y^2 + x y D[y, x] + (x^2 - x^4)(3y^2 + 3/2 x y D[y, x]) + x^6 (12y^2 + 3/2 x y D[y, x]) - x^2 (y^3 + 3/2 x y^2 D[y, x]) + x^4 (5y^3 + 3/2 x y^2 D[y, x]) - x^6 (8y^3 + 3/2 x y^2 D[y, x]) + (-x^4 + x^6)(2y^4 + 1/2 x y^3 D[y, x]) + O[x]^(2nn+1); If[y1 == y, Break[]]; y = y1]; CoefficientList[y, x^2]] // Rest;

seq[20] (* Jean-Fran├žois Alcover, Oct 05 2018, after Gheorghe Coserea *)

PROG

(PARI)

seq(N) = {

  my(x='x+O('x^(2*N+1)), y=0, y1=0, n=1);

  while (n++,

  y1 = x^2 + x^4 + 2*x^6 - 3*x^2*y + x^4*(-y + 1/2*x*y') +

       -x^6*(8*y + 1/2*x*y') + y^2 + x*y*y' +

       (x^2 - x^4)*(3*y^2 + 3/2*x*y*y') + x^6*(12*y^2 + 3/2*x*y*y') +

       -x^2*(y^3 + 1/2*x*3*y^2*y') + x^4*(5*y^3 + 1/2*x*3*y^2*y') +

       -x^6*(8*y^3 + 1/2*x*3*y^2*y') + (-x^4+x^6)*(2*y^4 + 1/8*x*4*y^3*y');

  if (y1 == y, break); y=y1);

  vector(N, n, polcoeff(y, 2*n));

};

seq(20) \\ Gheorghe Coserea, Oct 17 2017

CROSSREFS

Cf. A001147, A000698, A005411, A005412, A005413, A005416, A049464.

Sequence in context: A055608 A038742 A282709 * A044264 A044645 A153703

Adjacent sequences:  A005411 A005412 A005413 * A005415 A005416 A005417

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Gheorghe Coserea, Oct 17 2017

STATUS

approved

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Last modified May 11 21:35 EDT 2021. Contains 343808 sequences. (Running on oeis4.)