A005363 - OEIS

%I M1867 #61 Sep 16 2024 06:41:01

%S 1,2,8,44,310,2606,25202,272582,3233738,41454272,567709144,8230728508,

%T 125413517530,1996446632130,33039704641922,566087847780250,

%U 10006446665899330,181938461947322284,3393890553702212368,64807885247524512668,1264344439859632559216

%N Hoggatt sequence with parameter d=5.

%C 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

%C 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

%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="/A005363/b005363.txt">Table of n, a(n) for n = 0..681</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="http://www.fq.math.ca/Scanned/27-2/fielder.pdf">On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles</a>, Fib. Quart., 27 (1989), 160-168.

%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 From _Richard L. Ollerton_, Sep 12 2006: (Start)

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

%F (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)

%F a(n) = S(5,n) where S(d,n) is defined in A005364. - _Sean A. Irvine_, May 29 2016

%F a(n) ~ 9 * 2^(5*n + 27) / (sqrt(5) * Pi^2 * n^12). - _Vaclav Kotesovec_, Apr 01 2021

%F a(n) = Sum_{k=0..n} A056941(n, k) (row sums of triangle A056941). - _G. C. Greubel_, Nov 14 2022

%p a := n -> hypergeom([-4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5], -1):

%p seq(simplify(a(n)), n=0..25); # _Peter Luschny_, Feb 18 2021

%p # 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

%p v5 := proc(n) local t1,t2,t3,t4,t5;

%p if n=0 then 1

%p elif n=1 then 2

%p elif n=2 then 8

%p else

%p t1 := (4+n)*(5+n)^2*(6+n)*(7+n)*(8+n)*(252+253*n+55*n^2);

%p t2 := 3*(4+n)*(5+n)*(141120+362152*n + 373054*n^2+192647*n^3+52441*n^4 +7161*n^5 +385*n^6);

%p t3 := n*(1-n)*(5738880+14311976*n+14466242*n^2+7579175*n^3 +2170343*n^4+322289*n^5 + 19415*n^6);

%p t4 := 32*(2-n)*(1-n)^2*n^2*(1+n)*(560+363*n+55*n^2);

%p t5 := t2*v5(n-1)-t3*v5(n-2)+t4*v5(n-3);

%p t5/t1;

%p fi; end;

%p [seq(v5(n), n=0..20)];

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

%o (Magma)

%o A056941:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..4]]) >;

%o A005363:= func< n | (&+[A056941(n,k): k in [0..n]]) >;

%o [A005363(n): n in [0..40]]; // _G. C. Greubel_, Nov 14 2022

%o (SageMath)

%o def A005363(n): return simplify(hypergeometric([-4-n, -3-n, -2-n, -1-n, -n],[2,3,4,5], -1))

%o [A005363(n) for n in range(51)] # _G. C. Greubel_, Nov 14 2022

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

%Y Cf. A056941.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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