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%I A005362 M1789 #60 Sep 16 2024 06:41:35
%S A005362 1,2,7,32,177,1122,7898,60398,494078,4274228,38763298,366039104,
%T A005362 3579512809,36091415154,373853631974,3966563630394,42997859838010,
%U A005362 475191259977060,5344193918791710,61066078557804360,707984385321707910,8318207051955884772,98936727936728464152
%N A005362 Hoggatt sequence with parameter d=4.
%C A005362 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
%C A005362 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
%D A005362 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.
%D A005362 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005362 Seiichi Manyama, <a href="/A005362/b005362.txt">Table of n, a(n) for n = 0..845</a>
%H A005362 J. W. Essam and A. J. Guttmann, <a href="https://doi.org/10.1103/PhysRevE.52.5849">Vicious walkers and directed polymer networks in general dimensions</a>, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).
%H A005362 D. C. Fielder, <a href="/A005362/a005362.pdf">Letter to N. J. A. Sloane, Jun 1988</a>
%H A005362 D. C. Fielder and C. O. Alford, <a href="http://www.fq.math.ca/Scanned/27-2/fielder.pdf">On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles</a>, Fib. Quart., 27 (1989), 160-168.
%H A005362 D. C. Fielder and C. O. Alford, <a href="/A000108/a000108_19.pdf">An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles</a>, 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)
%H A005362 Nick Hobson, <a href="/A005362/a005362.py.txt">Python program for this sequence</a>
%H A005362 Vaclav Kotesovec, <a href="/A005366/a005366.pdf">Calculation of the asymptotic formula for the sequence A005366</a>
%F A005362 From _Richard L. Ollerton_, Sep 12 2006: (Start)
%F A005362 a(n) = Hypergeometric4F3([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1).
%F A005362 (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)
%F A005362 a(n) = S(4,n) where S(d,n) is defined in A005364. - _Sean A. Irvine_, May 29 2016
%F A005362 a(n) ~ 3 * 2^(4*n + 29/2) / (Pi^(3/2) * n^(15/2)). - _Vaclav Kotesovec_, Apr 01 2021
%p A005362 a := n -> hypergeom([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1):
%p A005362 seq(simplify(a(n)), n=0..25); # _Peter Luschny_, Feb 18 2021
%t A005362 A005362[n_]:=HypergeometricPFQ[{-3-n,-2-n,-1-n,-n},{2,3,4},1] (* _Richard L. Ollerton_, Sep 12 2006 *)
%o A005362 (Magma)
%o A005362 A056940:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..3]]) >;
%o A005362 A005362:= func< n | (&+[A056940(n,k): k in [0..n]]) >;
%o A005362 [A005362(n): n in [0..30]]; // _G. C. Greubel_, Nov 14 2022
%o A005362 (SageMath)
%o A005362 def A005362(n): return simplify(hypergeometric([-3-n, -2-n, -1-n, -n],[2,3,4], 1))
%o A005362 [A005362(n) for n in range(41)] # _G. C. Greubel_, Nov 14 2022
%Y A005362 Cf. A005364, A005365, A005366, A056940, A116925.
%K A005362 nonn
%O A005362 0,2
%A A005362 _N. J. A. Sloane_, _Simon Plouffe_

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