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
Keywords
Examples
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).
Links
- Robert Israel, Table of n, a(n) for n = 0..1491
- Liam Ayres, Evan Bialo, Aidan Cook, Alwin Chen, Matteus Froese, Erica Liu, Maryam Mohammadi Yekta, Oliver Pechenik, and Benjamin Wong, An exceptional equinumerosity of lattice paths and Young tableaux, arXiv:2506.03116 [math.CO], 2025. See p. 6.
- Richard K. Guy, Christian Krattenthaler and Bruce E. Sagan, Lattice paths, reflections, & dimension-changing bijections, Ars Combin. 34 (1992), 3-15.
- Mathematics Stack Exchange, Number of paths from (0,0) to (n,k) where all four directions are allowed, using a specific number of steps.
Crossrefs
Cf. A110609.
Programs
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Magma
[(4*n+4)*(2*n+1)*Binomial(2*n, n)/(n+2): n in [0..25]]; // Vincenzo Librandi, Jan 09 2015
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Maple
seq((4*n+4)*(2*n+1)*binomial(2*n, n)/(n+2), n=0..30);
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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 *)
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