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Number of simple tensors with n external gluons.
(Formerly M2080)
1

%I M2080 #34 Nov 21 2022 03:50:52

%S 1,0,1,2,15,140,1915,33810,734545,18929960,564216345,19088149850,

%T 722508543295,30249199720740,1387823333771875,69238799231051450,

%U 3731906171773805025,216101966957781304400,13379538319131196637425,881962125004262056604850

%N Number of simple tensors with n external gluons.

%C See Fig. 26, p. 1549 in the Cvitanovic reference. - _Jonathan Vos Post_, Feb 20 2010

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A005415/b005415.txt">Table of n, a(n) for n = 0..368</a>

%H P. Cvitanovic, <a href="https://web.archive.org/web/20171110034147/http://www.nbi.dk/~predrag/papers/PRD14-76.pdf">Group theory for Feynman diagrams in non-Abelian gauge theories</a>, Phys. Rev. D14 (1976), 1536-1553.

%F 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

%F 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

%t 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}]];

%t Table[a[n], {n,0,40}] (* _G. C. Greubel_, Nov 19 2022 *)

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A005415

%o if (n==0): return 1

%o 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))

%o [a(n) for n in range(40)] # _G. C. Greubel_, Nov 19 2022

%Y Cf. A001813.

%K nonn,nice

%O 0,4

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Jun 17 2016