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 A253487 Number of lattice paths of 2*n+2 steps in the first quadrant from (0,0) to (n,n). 3
 2, 16, 90, 448, 2100, 9504, 42042, 183040, 787644, 3359200, 14226212, 59907456, 251100200, 1048380480, 4362680250, 18103127040, 74934688620, 309509877600, 1275964023180, 5251296336000, 21579247511640, 88555121603520, 362957071241700, 1485969577717248 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS Robert Israel, Table of n, a(n) for n = 0..1491 Mathematics Stack Exchange, Number of paths from (0,0) to (n,k) where all four directions are allowed, using a specific number of steps. Richard K. Guy, Christian Krattenthaler and Bruce E. Sagan, Lattice paths, reflections, & dimension-changing bijections, Ars Combin. 34 (1992), 3-15. FORMULA a(n) = (4*n+4)*(2*n+1)*binomial(2*n, n)/(n+2). a(n) = 2*(n+5)*(n+1)*a(n-1)/(n*(n+2)) + (8*n-4)*a(n-2)/(n+2). G.f.: 1/x^2 - (1-6*x+4*x^2)/((1-4*x)^(3/2)*x^2). 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. a(n) = 2*A110609(n+1). - Vincenzo Librandi, Jan 09 2015 EXAMPLE 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). MAPLE seq((4*n+4)*(2*n+1)*binomial(2*n, n)/(n+2), n=0..30); MATHEMATICA 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 *) PROG (Magma) [(4*n+4)*(2*n+1)*Binomial(2*n, n)/(n+2): n in [0..25]]; // Vincenzo Librandi, Jan 09 2015 CROSSREFS Cf. A110609. Sequence in context: A000431 A281982 A207595 * A207019 A207164 A208113 Adjacent sequences: A253484 A253485 A253486 * A253488 A253489 A253490 KEYWORD nonn AUTHOR Robert Israel, Jan 02 2015 STATUS approved

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