A049464 - OEIS

%I #60 Dec 22 2023 12:11:30

%S 1,1,1,7,63,729,10113,161935,2923135,58547761,1286468225,30747331223,

%T 793992877247,22031281255689,653827064820993,20670172958564127,

%U 693662602935500031,24632233419065156193,922938914156271368961

%N Number of n-photon quenched skeletons.

%H Gheorghe Coserea, <a href="/A049464/b049464.txt">Table of n, a(n) for n = 0..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 D. J. Broadhurst, <a href="https://arXiv.org/abs/hep-ph/9909336">Four-loop Dyson-Schwinger-Johnson anatomy</a>, arXiv:hep-ph/9909336, 1999.

%H Ali Assem Mahmoud, <a href="https://arxiv.org/abs/2009.12688">An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams</a>, arXiv:2009.12688 [math.CO], 2020.

%H Ali Assem Mahmoud, <a href="https://arxiv.org/abs/2011.04291">Chord Diagrams and the Asymptotic Analysis of QED-type Theories</a>, arXiv:2011.04291 [hep-th], 2020.

%H Ali Assem Mahmoud, <a href="https://doi.org/10.1063/5.0171074">An asymptotic expansion for the number of two-connected chord diagrams</a>, J. Math. Phys. (2023) Vol. 64, 122301. See Section V.

%H Luca G. Molinari and Nicola Manini, <a href="https://arxiv.org/abs/cond-mat/0512342">Enumeration of many-body skeleton diagrams</a>, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

%F Reference gives recurrence.

%F From _Gheorghe Coserea_, Oct 22 2017: (Start)

%F 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)

%F 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).

%F 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)

%F 0 = 2*x*y*deriv(y,x) + (1+x)*y^2 - (2*x+1)*y + x.

%F (End)

%t 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 *)

%o (PARI)

%o seq(N) = {

%o   my(s=Ser(concat(1, vector(N, n, (2*n)!/(2^n*n!)))), g=(1/s - 1/s^2)/x);

%o   Vec(1 - 1/subst(g, 'x, serreverse(x*g^2*s^2)));

%o };

%o concat(1, seq(19))

%o \\ test: y='x*Ser(seq(200)); 0==2*x*y*y' + (1+x)*y^2 - (2*x+1)*y + x

%o \\ _Gheorghe Coserea_, Oct 12 2017

%Y Cf. A001147, A005416, A286795.

%K nonn,nice,easy

%O 0,4

%A _N. J. A. Sloane_