A005363 Hoggatt sequence with parameter d=5. (Formerly M1867)

1, 2, 8, 44, 310, 2606, 25202, 272582, 3233738, 41454272, 567709144, 8230728508, 125413517530, 1996446632130, 33039704641922, 566087847780250, 10006446665899330, 181938461947322284, 3393890553702212368, 64807885247524512668, 1264344439859632559216

Offset

0,2

Comments

Let V be the vector representation of SL(5) (of dimension 5) and let E be the exterior algebra of V (of dimension 32). 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 5-vicious walkers (aka vicious 5-watermelons) - see Essam and Guttmann (1995). This is the 5-walker analog of A001181. - N. J. A. Sloane, Mar 27 2021

References

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).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..681

J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).

D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988

D. C. Fielder and C. O. Alford, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart., 27 (1989), 160-168.

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, 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)

Formula

From Richard L. Ollerton, Sep 12 2006: (Start)

a(n) = Hypergeometric5F4([-4-n, -3-n, -2-n, -1-n, -n], [2,3,4,5], -1).

(n+4)*(n+5)^2*(n+6)*(n+7)*(n+8)*(252 +253*n +55*n^2)*a(n) = 3*(n+4)*(n+5)*(141120 + 362152*n + 373054*n^2 + 192647*n^3 + 52441*n^4 + 7161*n^5 + 385*n^6)*a(n-1) + n*(n-1)*(5738880 + 14311976*n + 14466242*n^2 + 7579175*n^3 + 2170343*n^4 + 322289*n^5 + 19415*n^6)*a(n-2) - 32*(n-1)^2*n^2*(n-2)*(n+1)*(560 + 363*n + 55*n^2)*a(n-3); a(-1)=a(0)=1, a(1)=2. (End)

a(n) = S(5,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016

a(n) ~ 9 * 2^(5*n + 27) / (sqrt(5) * Pi^2 * n^12). - Vaclav Kotesovec, Apr 01 2021

a(n) = Sum_{k=0..n} A056941(n, k) (row sums of triangle A056941). - G. C. Greubel, Nov 14 2022

Maple

a := n -> hypergeom([-4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5], -1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 18 2021
# The following Maple program is based on Eq (60) of Essam-Guttmann (1995) and confirms that that sequence is the same as the present one. - N. J. A. Sloane, Mar 27 2021
v5 := proc(n) local t1, t2, t3, t4, t5;
if n=0 then 1
elif n=1 then 2
elif n=2 then 8
else
t1 := (4+n)*(5+n)^2*(6+n)*(7+n)*(8+n)*(252+253*n+55*n^2);
t2 := 3*(4+n)*(5+n)*(141120+362152*n + 373054*n^2+192647*n^3+52441*n^4 +7161*n^5 +385*n^6);
t3 := n*(1-n)*(5738880+14311976*n+14466242*n^2+7579175*n^3 +2170343*n^4+322289*n^5 + 19415*n^6);
t4 := 32*(2-n)*(1-n)^2*n^2*(1+n)*(560+363*n+55*n^2);
t5 := t2*v5(n-1)-t3*v5(n-2)+t4*v5(n-3);
t5/t1;
fi; end;
[seq(v5(n), n=0..20)];

Mathematica

A005363[n_]:=HypergeometricPFQ[{-4-n, -3-n, -2-n, -1-n, -n}, {2, 3, 4, 5}, -1] (* Richard L. Ollerton, Sep 12 2006 *)

Prog

(Magma)
A056941:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..4]]) >;
A005363:= func< n | (&+[A056941(n, k): k in [0..n]]) >;
[A005363(n): n in [0..40]]; // G. C. Greubel, Nov 14 2022
(SageMath)
def A005363(n): return simplify(hypergeometric([-4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5], -1))
[A005363(n) for n in range(51)] # G. C. Greubel, Nov 14 2022

Crossrefs

Cf. A000079, A000108, A001181, A005362, A005364, A005365, A005366, A116925.

Cf. A056941.

Sequence in context: A253950 A212913 A321628 * A216234 A123307 A293905

Adjacent sequences: A005360 A005361 A005362 * A005364 A005365 A005366

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Sean A. Irvine, May 29 2016

Status

approved