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A005415 Number of simple tensors with n external gluons.
(Formerly M2080)
1
1, 0, 1, 2, 15, 140, 1915, 33810, 734545, 18929960, 564216345, 19088149850, 722508543295, 30249199720740, 1387823333771875, 69238799231051450, 3731906171773805025, 216101966957781304400, 13379538319131196637425, 881962125004262056604850 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
See Fig. 26, p. 1549 in the Cvitanovic reference. - Jonathan Vos Post, Feb 20 2010
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.
FORMULA
a(n) = Sum_{k=0..n-1} binomial(n-1, k) * a(k) * b(n-k) where b(1) = 0, b(2) = 1, b(n) = 2^(n-2) * (2*n-5)!! = A001813(n-2) [from Cvitanovic]. - Sean A. Irvine, Jun 17 2016
a(n) = Sum_{k=0..n-2} binomial(n-1, k) * ((2*n-2*k-4)!/(n-k-2)!) * a(k), with a(0) = 1. - G. C. Greubel, Nov 19 2022
MATHEMATICA
a[n_]:= a[n]= If[n==0, 1, Sum[Binomial[n-1, k]*((2*n-2*k-4)!/(n-k-2)!)*a[k], {k, 0, n-2}]];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Nov 19 2022 *)
PROG
(SageMath)
@CachedFunction
def a(n): # a = A005415
if (n==0): return 1
else: return sum(binomial(n-1, k)*factorial(n-k-2)*binomial(2*n-2*k-4, n-k-2)*a(k) for k in (0..n-2))
[a(n) for n in range(40)] # G. C. Greubel, Nov 19 2022
CROSSREFS
Cf. A001813.
Sequence in context: A285449 A371584 A288950 * A371523 A219868 A224885
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Jun 17 2016
STATUS
approved

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Last modified June 23 21:27 EDT 2024. Contains 373654 sequences. (Running on oeis4.)