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A005366 Hoggatt sequence with parameter d=8.

%I A005366 M2006 #73 Sep 15 2024 17:15:32
%S A005366 1,2,11,92,1157,19142,403691,10312304,311348897,10826298914,
%T A005366 426196716090,18700516849302,903666922873158,47592378143008974,
%U A005366 2708388575679431454,165309083872549538190,10753269337589887334670,741379205762167719365268
%N A005366 Hoggatt sequence with parameter d=8.
%C A005366 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
%C A005366 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
%C A005366 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
%D A005366 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 A005366 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005366 Seiichi Manyama, <a href="/A005366/b005366.txt">Table of n, a(n) for n = 0..438</a>
%H A005366 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 A005366 D. C. Fielder, <a href="/A005362/a005362.pdf">Letter to N. J. A. Sloane, Jun 1988</a>
%H A005366 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 A005366 Vaclav Kotesovec, <a href="/A005366/a005366.pdf">Calculation of the asymptotic formula for the sequence A005366</a>
%F A005366 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
%F A005366 a(n) = S(8,n) where S(d,n) is defined in A005364. - _Sean A. Irvine_, May 29 2016
%F A005366 a(n) ~ 1913625 * 2^(8*n + 74) / (Pi^(7/2) * n^(63/2)). - _Vaclav Kotesovec_, Apr 01 2021
%t A005366 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 *)
%o A005366 (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
%o A005366 (Magma)
%o A005366 A142468:= func< n,k | Binomial(n,k)*(&*[Binomial(n+2*j,k+j)/Binomial(n+2*j,j): j in [1..7]]) >;
%o A005366 A005366:= func< n | (&+[A142468(n,k): k in [0..n]]) >;
%o A005366 [A005366(n): n in [0..40]]; // _G. C. Greubel_, Nov 13 2022
%o A005366 (SageMath)
%o A005366 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))
%o A005366 [A005365(n) for n in range(51)] # _G. C. Greubel_, Nov 13 2022
%Y A005366 Row sums of A142468.
%Y A005366 Cf. A000079, A000108, A001181, A005362, A005363, A005364, A005365, A116925.
%K A005366 nonn
%O A005366 0,2
%A A005366 _N. J. A. Sloane_
%E A005366 More terms from _Sean A. Irvine_, May 29 2016