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A051862 Perturbation expansion in quantum field theory: scalar case in 6 spacetime dimensions. 2
0, 1, 11, 376, 20241, 1427156, 121639250, 12007003824, 1337583507153, 165328009728652, 22404009743110566, 3299256277254713760, 524366465815117346250, 89448728780073829991976, 16301356287284530869810308, 3161258841758986060906197536, 650090787950164885954804021185 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

D. J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Padé-Borel resummation. Phys. Lett. B 475 (2000), 63-70.

LINKS

Table of n, a(n) for n=0..16.

D. J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Padé-Borel resummation, arXiv:hep-th/9912093, 1999-2000.

FORMULA

The generating procedure is described by Broadhurst and Kreimer.

EXAMPLE

a(31) = 7632236320181399967333968684399053053157812979126909028545984868160 was computed using Kreimer's Hopf algebra of rooted trees. It subsumes 2.6*10^21 terms in quantum field theory.

PROG

(Sage)

t = PowerSeriesRing(QQ, 't').gen()

def shadok(c):

    """

    fixed point procedure after G. Dunne talk at Kreimer's fest 2020

    """

    aa_sur_c = 2 * t * c.derivative() - c - 3

    aa = c * aa_sur_c

    bb_sur_c = 2 * t * aa.derivative() - aa - 2 * aa_sur_c

    bb = c * bb_sur_c

    cc_sur_c = 2 * t * bb.derivative() - bb - bb_sur_c

    return 3 * t / cc_sur_c

C = (-t / 2).O(2)

for k in range(10):

    C = shadok(C)

list(1 / 6 * C(-12 * t))

# F. Chapoton, Nov 19 2020

CROSSREFS

Cf. A000699.

Sequence in context: A024149 A333466 A018893 * A006698 A048431 A286914

Adjacent sequences:  A051859 A051860 A051861 * A051863 A051864 A051865

KEYWORD

nonn

AUTHOR

David Broadhurst, Dec 14 1999

EXTENSIONS

a(0)=0 and more terms from F. Chapoton, Nov 19 2020

STATUS

approved

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Last modified April 11 00:03 EDT 2021. Contains 342877 sequences. (Running on oeis4.)