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%I #21 Jun 16 2021 20:59:44
%S 0,1,11,376,20241,1427156,121639250,12007003824,1337583507153,
%T 165328009728652,22404009743110566,3299256277254713760,
%U 524366465815117346250,89448728780073829991976,16301356287284530869810308,3161258841758986060906197536,650090787950164885954804021185
%N Perturbation expansion in quantum field theory: scalar case in 6 spacetime dimensions.
%H Michael Borinsky, Gerald V. Dunne, and Max Meynig, <a href="https://arxiv.org/abs/2104.00593">Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: phi^3 QFT in 6 Dimensions</a>, arXiv:2104.00593 [hep-th], 2021.
%H D. J. Broadhurst and D. Kreimer, <a href="http://arXiv.org/abs/hep-th/9912093">Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Padé-Borel resummation</a>, arXiv:hep-th/9912093, 1999-2000. See also <a href="https://www.infona.pl/resource/bwmeta1.element.elsevier-ff9aba33-6f7c-3ed7-be8d-c0263813d125/tab/summary">Phys. Lett. B</a> (2000) Vol. 475, 63-70.
%F The generating procedure is described by Broadhurst and Kreimer.
%e a(31) = 7632236320181399967333968684399053053157812979126909028545984868160 was computed using Kreimer's Hopf algebra of rooted trees. It subsumes 2.6*10^21 terms in quantum field theory.
%o (Sage)
%o t = PowerSeriesRing(QQ, 't').gen()
%o def shadok(c):
%o """
%o fixed point procedure after G. Dunne talk at Kreimer's fest 2020
%o """
%o aa_sur_c = 2 * t * c.derivative() - c - 3
%o aa = c * aa_sur_c
%o bb_sur_c = 2 * t * aa.derivative() - aa - 2 * aa_sur_c
%o bb = c * bb_sur_c
%o cc_sur_c = 2 * t * bb.derivative() - bb - bb_sur_c
%o return 3 * t / cc_sur_c
%o C = (-t / 2).O(2)
%o for k in range(10):
%o C = shadok(C)
%o list(1 / 6 * C(-12 * t))
%o # _F. Chapoton_, Nov 19 2020
%Y Cf. A000699.
%K nonn
%O 0,3
%A _David Broadhurst_, Dec 14 1999
%E a(0)=0 and more terms from _F. Chapoton_, Nov 19 2020
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