A005364 Hoggatt sequence with parameter d=6. (Formerly M1943)

1, 2, 9, 58, 506, 5462, 70226, 1038578, 17274974, 317292692, 6346909285, 136723993122, 3143278648954, 76547029418394, 1962350550273130, 52679691605422354, 1474290522744355250, 42847373913958703100, 1288899422418558314550, 40013380588722843337620

Offset

0,2

Comments

Let V be the vector representation of SL(6) (of dimension 6) and let E be the exterior algebra of V (of dimension 64). 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 6-vicious walkers (aka vicious 6-watermelons) - see Essam and Guttmann (1995). This is the 6-walker analog of A001181. - N. J. A. Sloane, Mar 27 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..573

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)

Formula

a(n) = Hypergeometric6F5([-5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6], 1). - Richard L. Ollerton, Sep 13 2006

a(n) = S(6,n) where S(d,n) = 1 + Sum_{h=0..n-1} Product_{k=0..h} binomial(n+d-1-k,d) / binomial(d + k, d) [From Fielder and Alford]. - Sean A. Irvine, May 29 2016

a(n) ~ 135 * 2^(6*n + 40) / (sqrt(3) * Pi^(5/2) * n^(35/2)). - Vaclav Kotesovec, Apr 01 2021

Mathematica

A005364[n_]:=HypergeometricPFQ[{-5-n, -4-n, -3-n, -2-n, -1-n, -n}, {2, 3, 4, 5, 6}, 1] (* Richard L. Ollerton, Sep 13 2006 *)

Prog

(PARI) a(n) = my(d=6); 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)
A142465:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..5]]) >;
A005364:= func< n | (&+[A142465(n, k): k in [0..n]]) >;
[A005364(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022
(SageMath)
def A005364(n): return simplify(hypergeometric([-5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6], 1))
[A005364(n) for n in range(51)] # G. C. Greubel, Nov 13 2022

Crossrefs

Row sums of A142465.

Cf. A000079, A000108, A001181, A005362, A005363, A005365, A005366, A116925.

Sequence in context: A277358 A156129 A191811 * A267464 A184355 A346519

Adjacent sequences: A005361 A005362 A005363 * A005365 A005366 A005367

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, May 29 2016

Status

approved