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A005559
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Number of walks on square lattice.
(Formerly M1832)
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4
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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
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OFFSET
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1,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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