# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a005559 Showing 1-1 of 1 %I A005559 M1832 #55 Jun 24 2023 12:47:32 %S A005559 1,2,8,20,75,210,784,2352,8820,27720,104544,339768,1288287,4294290, %T A005559 16359200,55621280,212751396,734959368,2821056160,9873696560, %U A005559 38013731756,134510127752,519227905728,1854385377600,7174705330000,25828939188000,100136810390400 %N A005559 Number of walks on square lattice. %D A005559 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A005559 R. K. Guy, Letter to N. J. A. Sloane, May 1990 %H A005559 R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 (see figure 5). %F A005559 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 %F A005559 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 %F A005559 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 %p A005559 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 %t A005559 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 *) %o A005559 (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))} %o A005559 (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 %Y A005559 Cf. A005558-A005560, A093768. %K A005559 nonn,walk %O A005559 1,2 %A A005559 _N. J. A. Sloane_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE