A005415 Number of simple tensors with n external gluons. (Formerly M2080)

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

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

Cf. A001813.

Sequence in context: A285449 A371584 A288950 * A371523 A219868 A224885

Adjacent sequences: A005412 A005413 A005414 * A005416 A005417 A005418

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Jun 17 2016

Status

approved