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%I M1832 #55 Jun 24 2023 12:47:32
%S 1,2,8,20,75,210,784,2352,8820,27720,104544,339768,1288287,4294290,
%T 16359200,55621280,212751396,734959368,2821056160,9873696560,
%U 38013731756,134510127752,519227905728,1854385377600,7174705330000,25828939188000,100136810390400
%N Number of walks on square lattice.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>
%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 (see figure 5).
%F 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 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 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 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 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 (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 (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 Cf. A005558-A005560, A093768.
%K nonn,walk
%O 1,2
%A _N. J. A. Sloane_
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