Voozh

A005415

Number of simple tensors with n external gluons.

1, 0, 1, 2, 15, 140, 1915, 33810, 734545, 18929960, 564216345, 19088149850, 722508543295, 30249199720740, 1387823333771875, 69238799231051450, 3731906171773805025, 216101966957781304400, 13379538319131196637425, 881962125004262056604850

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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

Alois P. Heinz, Table of n, a(n) for n = 0..368

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