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%I M1976 #66 Nov 14 2022 00:41:19
%S 1,2,10,74,782,10562,175826,3457742,78408332,2005691690,56970282514,
%T 1772967273794,59814500606018,2168062920325850,83802728579860658,
%U 3432438439271783026,148165335791410936770,6708873999658599592672
%N Hoggatt sequence with parameter d=7.
%C Let V be the vector representation of SL(7) (of dimension 7) and let E be the exterior algebra of V (of dimension 128). 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 This is the number of 7-vicious walkers (aka vicious 7-watermelons) - see Essam and Guttmann (1995). This is the 7-walker analog of A001181. - _N. J. A. Sloane_, Mar 27 2021
%D 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A005365/b005365.txt">Table of n, a(n) for n = 0..496</a>
%H 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 D. C. Fielder, <a href="/A005362/a005362.pdf">Letter to N. J. A. Sloane, Jun 1988</a>
%H 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)
%F a(n) = Hypergeometric7F6([-6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6, 7], -1). - _Richard L. Ollerton_, Sep 13 2006
%F a(n) = S(7,n) where S(d,n) is defined in A005364. - _Sean A. Irvine_, May 29 2016
%F a(n) ~ 6075 * 2^(7*n + 57) / (sqrt(7) * Pi^3 * n^24). - _Vaclav Kotesovec_, Apr 01 2021
%t A005365[n_]:=HypergeometricPFQ[{-6-n,-5-n,-4-n,-3-n,-2-n,-1-n,-n},{2,3,4,5,6,7},-1] (* _Richard L. Ollerton_, Sep 13 2006 *)
%o (PARI) a(n) = my(d=7); 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 (Magma)
%o A142467:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..6]]) >;
%o A005365:= func< n | (&+[A142467(n,k): k in [0..n]]) >;
%o [A005365(n): n in [0..40]]; // _G. C. Greubel_, Nov 13 2022
%o (SageMath)
%o def A005365(n): return simplify(hypergeometric([-6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n], [2,3,4,5,6,7], -1))
%o [A005365(n) for n in range(51)] # _G. C. Greubel_, Nov 13 2022
%Y Row sums of A142467.
%Y Cf. A000079, A000108, A001181, A005362, A005363, A005364, A005366, A116925.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, May 29 2016
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