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%I #14 Sep 08 2022 08:46:10
%S 2,16,90,448,2100,9504,42042,183040,787644,3359200,14226212,59907456,
%T 251100200,1048380480,4362680250,18103127040,74934688620,309509877600,
%U 1275964023180,5251296336000,21579247511640,88555121603520,362957071241700,1485969577717248
%N Number of lattice paths of 2*n+2 steps in the first quadrant from (0,0) to (n,n).
%H Robert Israel, <a href="/A253487/b253487.txt">Table of n, a(n) for n = 0..1491</a>
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/870446/number-of-paths-from-0-0-to-n-k-where-all-four-directions-are-allowed-u/870456#comment2216962_870456">Number of paths from (0,0) to (n,k) where all four directions are allowed, using a specific number of steps</a>.
%H Richard K. Guy, Christian Krattenthaler and Bruce E. Sagan, <a href="http://www.mat.univie.ac.at/~kratt/artikel/paths.html">Lattice paths, reflections, & dimension-changing bijections</a>, Ars Combin. 34 (1992), 3-15.
%F a(n) = (4*n+4)*(2*n+1)*binomial(2*n, n)/(n+2).
%F a(n) = 2*(n+5)*(n+1)*a(n-1)/(n*(n+2)) + (8*n-4)*a(n-2)/(n+2).
%F G.f.: 1/x^2 - (1-6*x+4*x^2)/((1-4*x)^(3/2)*x^2).
%F E.g.f.: 16*x*exp(2*x)*I_0(2*x) + (2-4*x+16*x^2)*exp(2*x)*I_1(2*x)/x where I_0, I_1 are modified Bessel functions.
%F a(n) = 2*A110609(n+1). - _Vincenzo Librandi_, Jan 09 2015
%e For n = 0 the a(0) = 2 paths of length 2 from (0,0) to (0,0) are (0,0)->(1,0)->(0,0) and (0,0)->(0,1)->(0,0).
%p seq((4*n+4)*(2*n+1)*binomial(2*n, n)/(n+2), n=0..30);
%t Table[(4 n + 4) (2 n + 1) Binomial[2 n, n] / (n + 2), {n, 0, 25}] (* or *) CoefficientList[Series[1 / x^2 - (1 - 6 x + 4 x^2) / ((1 - 4 x)^(3/2) x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jan 09 2015 *)
%o (Magma) [(4*n+4)*(2*n+1)*Binomial(2*n, n)/(n+2): n in [0..25]]; // _Vincenzo Librandi_, Jan 09 2015
%Y Cf. A110609.
%K nonn
%O 0,1
%A _Robert Israel_, Jan 02 2015
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