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A049464 Number of n-photon quenched skeletons. 9
1, 1, 1, 7, 63, 729, 10113, 161935, 2923135, 58547761, 1286468225, 30747331223, 793992877247, 22031281255689, 653827064820993, 20670172958564127, 693662602935500031, 24632233419065156193, 922938914156271368961 (list; graph; refs; listen; history; text; internal format)
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.

Luca G. Molinari, 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-Fran├žois 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

Cf. A001147, A005416, A286795.

Sequence in context: A051579 A185106 A275577 * A229078 A084063 A184141

Adjacent sequences:  A049461 A049462 A049463 * A049465 A049466 A049467

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 23 03:25 EDT 2019. Contains 325230 sequences. (Running on oeis4.)