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Hoggatt sequence with parameter d=6.
(Formerly M1943)
7

%I M1943 #70 Sep 16 2024 06:40:30

%S 1,2,9,58,506,5462,70226,1038578,17274974,317292692,6346909285,

%T 136723993122,3143278648954,76547029418394,1962350550273130,

%U 52679691605422354,1474290522744355250,42847373913958703100,1288899422418558314550,40013380588722843337620

%N Hoggatt sequence with parameter d=6.

%C 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

%C 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

%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="/A005364/b005364.txt">Table of n, a(n) for n = 0..573</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)

%H Vaclav Kotesovec, <a href="/A005366/a005366.pdf">Calculation of the asymptotic formula for the sequence A005366</a>

%F 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

%F 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

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

%t 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 *)

%o (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

%o (Magma)

%o A142465:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..5]]) >;

%o A005364:= func< n | (&+[A142465(n,k): k in [0..n]]) >;

%o [A005364(n): n in [0..40]]; // _G. C. Greubel_, Nov 13 2022

%o (SageMath)

%o def A005364(n): return simplify(hypergeometric([-5-n, -4-n, -3-n, -2-n, -1-n, -n],[2, 3, 4, 5, 6], 1))

%o [A005364(n) for n in range(51)] # _G. C. Greubel_, Nov 13 2022

%Y Row sums of A142465.

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

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, May 29 2016