A005559 Number of walks on square lattice. (Formerly M1832)

1, 2, 8, 20, 75, 210, 784, 2352, 8820, 27720, 104544, 339768, 1288287, 4294290, 16359200, 55621280, 212751396, 734959368, 2821056160, 9873696560, 38013731756, 134510127752, 519227905728, 1854385377600, 7174705330000, 25828939188000, 100136810390400

Offset

1,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..27.

R. K. Guy, Letter to N. J. A. Sloane, May 1990

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 (see figure 5).

Formula

a(n) = C(n+1,ceiling((n-1)/2)) *C(n,floor((n-1)/2)) -C(n+1,ceiling((n-2)/2)) *C(n,floor((n-2)/2)). - Paul D. Hanna, Apr 16 2004

G.f.: -(48*x^3-16*x^2-3*x+1)*EllipticK(4*x)/(12*Pi*x^4)+(4*x^2-9*x+1)*EllipticE(4*x)/(12*Pi*x^4)+1/(4*x^3)-1/(2*x^2) (using Maple's convention for elliptic integrals: EllipticE(t) = Integral_{s=0..1} sqrt(1 - s^2*t^2)/sqrt(1-s^2) ds, EllipticK(t) = Integral_{s=0..1} ((1-s^2*t^2)*(1-s^2))^(-1/2) ds). - Robert Israel, Oct 19 2014

Conjecture: -(n-1)*(2*n+1)*(n+4)*(n+3)*a(n) +4*(n+1)*(2*n^2+4*n+9)*a(n-1) +16*n*(n-1)*(2*n+3)*(n+1)*a(n-2)=0. - R. J. Mathar, Apr 02 2017

Maple

seq(binomial(n+1, ceil((n-1)/2))*binomial(n, floor((n-1)/2)) -binomial(n+1, ceil((n-2)/2))*binomial(n, floor((n-2)/2)), n=1..30); # Robert Israel, Oct 19 2014

Mathematica

Table[Binomial[n+2, Ceiling[n/2]] Binomial[n+1, Floor[n/2]] - Binomial[n+2, Ceiling[(n-1)/2]] Binomial[n+1, Floor[(n-1)/2]], {n, 0, 200}] (* Vincenzo Librandi, Oct 17 2014 *)

Prog

(PARI) {a(n)=binomial(n+2, ceil(n/2))*binomial(n+1, floor(n/2)) - binomial(n+2, ceil((n-1)/2))*binomial(n+1, floor((n-1)/2))}
(Magma) [Binomial(n+2, Ceiling(n/2))*Binomial(n+1, Floor(n/2)) - Binomial(n+2, Ceiling((n-1)/2))*Binomial(n+1, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

Crossrefs

Cf. A005558-A005560, A093768.

Sequence in context: A100097 A133467 A091004 * A001471 A162585 A000159

Adjacent sequences: A005556 A005557 A005558 * A005560 A005561 A005562

Keyword

nonn,walk

Author

N. J. A. Sloane

Status

approved