A005366 Hoggatt sequence with parameter d=8.

1, 2, 11, 92, 1157, 19142, 403691, 10312304, 311348897, 10826298914, 426196716090, 18700516849302, 903666922873158, 47592378143008974, 2708388575679431454, 165309083872549538190, 10753269337589887334670, 741379205762167719365268

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

Row sums of A142468.

Cf. A000079, A000108, A001181, A005362, A005363, A005364, A005365, A116925.

Programs

Magma

A142468:= func< n,k | Binomial(n,k)*(&*[Binomial(n+2*j,k+j)/Binomial(n+2*j,j): j in [1..7]]) >;
A005366:= func< n | (&+[A142468(n,k): k in [0..n]]) >;
[A005366(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022

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 *)

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

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

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

Extensions

More terms from Sean A. Irvine, May 29 2016