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A005362
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Hoggatt sequence with parameter d=4.
(Formerly M1789)
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8
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1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, 366039104, 3579512809, 36091415154, 373853631974, 3966563630394, 42997859838010, 475191259977060, 5344193918791710, 61066078557804360, 707984385321707910, 8318207051955884772, 98936727936728464152
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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) ~ 3 * 2^(4*n + 29/2) / (Pi^(3/2) * n^(15/2)). - Vaclav Kotesovec, Apr 01 2021
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MAPLE
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a := n -> hypergeom([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1):
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MATHEMATICA
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PROG
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(Magma)
A056940:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..3]]) >;
(SageMath)
def A005362(n): return simplify(hypergeometric([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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