OFFSET
0,2
COMMENTS
Let V be the vector representation of SL(8) (of dimension 8) and let E be the exterior algebra of V (of dimension 256). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 03 2021
This is the number of 8-vicious walkers (aka vicious 8-watermelons) - see Essam and Guttmann (1995). This is the 8-walker analog of A001181. - N. J. A. Sloane, Mar 27 2021
In general, for d > 0, a(n) ~ BarnesG(d+1) * 2^(d*n + (2*d+1)*(d-1)/2) / (sqrt(d) * Pi^((d-1)/2) * n^((d^2 - 1)/2)). - Vaclav Kotesovec, Apr 01 2021
REFERENCES
D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt sums and Hoggatt triangles, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers: Proc. Third Internat. Conf. on Fibonacci Numbers and Their Applications, Pisa, Jul 25-29, 1988. Kluwer, Dordrecht, Vol. 3, 1990, pp. 77-88.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..438
J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).
D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988
D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)
Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366
FORMULA
a(n) = Hypergeometric8F7([-7-n, -6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n],[2, 3, 4, 5, 6, 7, 8], 1). - Richard L. Ollerton, Sep 13 2006
a(n) = S(8,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016
a(n) ~ 1913625 * 2^(8*n + 74) / (Pi^(7/2) * n^(63/2)). - Vaclav Kotesovec, Apr 01 2021
MATHEMATICA
A005366[n_]:=HypergeometricPFQ[{-7-n, -6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n}, {2, 3, 4, 5, 6, 7, 8}, 1] (* Richard L. Ollerton, Sep 13 2006 *)
PROG
(PARI) a(n) = my(d=8); 1 + sum(h=0, n-1, prod(k=0, h, binomial(n+d-1-k, d) / binomial(d + k, d))); \\ Michel Marcus, Feb 08 2021
(Magma)
A142468:= func< n, k | Binomial(n, k)*(&*[Binomial(n+2*j, k+j)/Binomial(n+2*j, j): j in [1..7]]) >;
[A005366(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022
(SageMath)
def A005365(n): return simplify(hypergeometric([-7-n, -6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6, 7, 8], 1))
[A005365(n) for n in range(51)] # G. C. Greubel, Nov 13 2022
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, May 29 2016
STATUS
approved