A005362 Hoggatt sequence with parameter d=4. (Formerly M1789)

1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, 366039104, 3579512809, 36091415154, 373853631974, 3966563630394, 42997859838010, 475191259977060, 5344193918791710, 61066078557804360, 707984385321707910, 8318207051955884772, 98936727936728464152

Offset

0,2

Comments

Let V be the vector representation of SL(4) (of dimension 4) and let E be the exterior algebra of V (of dimension 16). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 18 2021

This is the number of 4-vicious walkers (aka vicious 4-watermelons) - see Essam and Guttmann (1995). This is the 4-walker analog of A001181. - N. J. A. Sloane, Mar 22 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..845

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, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart., 27 (1989), 160-168.

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)

Nick Hobson, Python program for this sequence

Formula

From Richard L. Ollerton, Sep 12 2006: (Start)

a(n) = Hypergeometric4F3([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1).

(n+3)*(n+4)*(n+5)*(n+6)*a(n) = 6*(n+1)*(n+3)*(n+4)*(2*n+5)*a(n-1) + 4*(n-1)*n*(4*n+7)*(4*n+9)*a(n-2); a(0)=1, a(1)=2. (End)

a(n) = S(4,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016

a(n) ~ 3 * 2^(4*n + 29/2) / (Pi^(3/2) * n^(15/2)). - Vaclav Kotesovec, Apr 01 2021

Maple

a := n -> hypergeom([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 18 2021

Mathematica

A005362[n_]:=HypergeometricPFQ[{-3-n, -2-n, -1-n, -n}, {2, 3, 4}, 1] (* Richard L. Ollerton, Sep 12 2006 *)

Prog

(Magma)
A056940:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..3]]) >;
A005362:= func< n | (&+[A056940(n, k): k in [0..n]]) >;
[A005362(n): n in [0..30]]; // G. C. Greubel, Nov 14 2022
(SageMath)
def A005362(n): return simplify(hypergeometric([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1))
[A005362(n) for n in range(41)] # G. C. Greubel, Nov 14 2022

Crossrefs

Cf. A005364, A005365, A005366, A056940, A116925.

Sequence in context: A125223 A339226 A277359 * A059439 A190123 A006014

Adjacent sequences: A005359 A005360 A005361 * A005363 A005364 A005365

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Status

approved