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A005414 Feynman diagrams of order 2n with vertex skeletons.

%I A005414 M4906 #44 Jul 08 2025 16:31:02
%S A005414 1,1,13,93,1245,18093,308605,5887453,124221373,2864305277,71589605885,
%T A005414 1927010749181,55572839581437,1709604517055229,55893262628149245,
%U A005414 1935654236127347709,70799043456576835581,2727771901780930132989,110438840436968476274685,4688223534904569925386237
%N A005414 Feynman diagrams of order 2n with vertex skeletons.
%D A005414 P. Cvitanovic, B. Lautrup and R. B. Pearson, Number and weights of Feynman diagrams, Phys. Rev. D 18 (1978), 1939-1949.
%D A005414 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005414 Gheorghe Coserea, <a href="/A005414/b005414.txt">Table of n, a(n) for n = 1..202</a>
%H A005414 Michael Borinsky, <a href="https://arxiv.org/abs/1703.00840">Renormalized asymptotic enumeration of Feynman diagrams</a>, arXiv:1703.00840 [hep-th], 2017.
%H A005414 P. Cvitanovic, B. Lautrup and R. B. Pearson, <a href="http://www.nbi.dk/~predrag/papers/PRD18-78.pdf">The number and weights of Feynman diagrams</a>, Phys. Rev. D18, 1939 (1978).
%F A005414 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
%t A005414 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;
%t A005414 seq[20] (* _Jean-François Alcover_, Oct 05 2018, after _Gheorghe Coserea_ *)
%o A005414 (PARI)
%o A005414 seq(N) = {
%o A005414   my(x='x+O('x^(2*N+1)), y=0, y1=0, n=1);
%o A005414   while (n++,
%o A005414   y1 = x^2 + x^4 + 2*x^6 - 3*x^2*y + x^4*(-y + 1/2*x*y') +
%o A005414        -x^6*(8*y + 1/2*x*y') + y^2 + x*y*y' +
%o A005414        (x^2 - x^4)*(3*y^2 + 3/2*x*y*y') + x^6*(12*y^2 + 3/2*x*y*y') +
%o A005414        -x^2*(y^3 + 1/2*x*3*y^2*y') + x^4*(5*y^3 + 1/2*x*3*y^2*y') +
%o A005414        -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');
%o A005414   if (y1 == y, break); y=y1);
%o A005414   vector(N, n, polcoeff(y, 2*n));
%o A005414 };
%o A005414 seq(20) \\ _Gheorghe Coserea_, Oct 17 2017
%Y A005414 Cf. A001147, A000698, A005411, A005412, A005413, A005416, A049464.
%K A005414 nonn
%O A005414 1,3
%A A005414 _N. J. A. Sloane_
%E A005414 More terms from _Gheorghe Coserea_, Oct 17 2017