A253487 Number of lattice paths of 2*n+2 steps in the first quadrant from (0,0) to (n,n).

2, 16, 90, 448, 2100, 9504, 42042, 183040, 787644, 3359200, 14226212, 59907456, 251100200, 1048380480, 4362680250, 18103127040, 74934688620, 309509877600, 1275964023180, 5251296336000, 21579247511640, 88555121603520, 362957071241700, 1485969577717248

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