OFFSET
0,4
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..202
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
D. J. Broadhurst, Four-loop Dyson-Schwinger-Johnson anatomy, arXiv:hep-ph/9909336, 1999.
Ali Assem Mahmoud, An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams, arXiv:2009.12688 [math.CO], 2020.
Ali Assem Mahmoud, Chord Diagrams and the Asymptotic Analysis of QED-type Theories, arXiv:2011.04291 [hep-th], 2020.
Ali Assem Mahmoud, An asymptotic expansion for the number of two-connected chord diagrams, J. Math. Phys. (2023) Vol. 64, 122301. See Section V.
Luca G. Molinari and Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.
FORMULA
Reference gives recurrence.
From Gheorghe Coserea, Oct 22 2017: (Start)
a(n) ~ 2*exp(-2)/sqrt(Pi) * n^(1/2) * 2^n * n! * (1 - 21/(8*n) - 87/(128*n^2) + O(1/n^3)). (see Borinsky link)
For n > 0 we have a(n) == 1 (mod 8) if n mod 8 in {1,2,5,6}, otherwise a(n) == 7 (mod 8).
G.f. y(x) satisfies (with a(0)=0): g = 1 + g*y(x*g^2*s^2), where s = A001147(x) and g = A005416(x). (eqn. (7) in Broadhurst link)
0 = 2*x*y*deriv(y,x) + (1+x)*y^2 - (2*x+1)*y + x.
(End)
MATHEMATICA
terms = 19; y[_] = 0; Do[y[x_] = (x + (1 + x)*y[x]^2 + 2*x*y[x]*y'[x])/(1 + 2*x) + O[x]^terms // Normal, terms]; CoefficientList[1 + y[x], x] (* Jean-François Alcover, Aug 14 2013, updated Jan 12 2018 *)
PROG
(PARI)
seq(N) = {
my(s=Ser(concat(1, vector(N, n, (2*n)!/(2^n*n!)))), g=(1/s - 1/s^2)/x);
Vec(1 - 1/subst(g, 'x, serreverse(x*g^2*s^2)));
};
concat(1, seq(19))
\\ test: y='x*Ser(seq(200)); 0==2*x*y*y' + (1+x)*y^2 - (2*x+1)*y + x
\\ Gheorghe Coserea, Oct 12 2017
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved