A005414 - OEIS

%I M4906 #40 Oct 05 2018 04:16:21

%S 1,1,13,93,1245,18093,308605,5887453,124221373,2864305277,71589605885,

%T 1927010749181,55572839581437,1709604517055229,55893262628149245,

%U 1935654236127347709,70799043456576835581,2727771901780930132989,110438840436968476274685,4688223534904569925386237

%N Feynman diagrams of order 2n with vertex skeletons.

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

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

%H Gheorghe Coserea, <a href="/A005414/b005414.txt">Table of n, a(n) for n = 1..202</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 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 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 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 seq[20] (* _Jean-François Alcover_, Oct 05 2018, after _Gheorghe Coserea_ *)

%o (PARI)

%o seq(N) = {

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

%o   while (n++,

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

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

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

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

%o        -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   if (y1 == y, break); y=y1);

%o   vector(N, n, polcoeff(y, 2*n));

%o };

%o seq(20) \\ _Gheorghe Coserea_, Oct 17 2017

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

%K nonn

%O 1,3

%A _N. J. A. Sloane_.

%E More terms from _Gheorghe Coserea_, Oct 17 2017