OFFSET
0,1
COMMENTS
a(n) is the number of standard Young tableaux of shape (n+2,2,1^n). - Oliver Pechenik, Sep 18 2025
LINKS
Robert Israel, Table of n, a(n) for n = 0..1491
Liam Ayres, Evan Bialo, Alwin Chen, Aidan Cook, Matteus Froese, Erica Liu, Maryam Mohammadi Yekta, Oliver Pechenik, and Benjamin Wong, An exceptional equinumerosity of lattice paths and Young tableaux, Journal of Integer Sequences 28 (2025), Article 25.5.2; also on ArXiv, arXiv:2506.03116 [math.CO], 2025.
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.
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
From Amiram Eldar, Sep 25 2025: (Start)
a(n) ~ 2^(2*n+3) * sqrt(n/Pi).
Sum_{n>=0} 1/a(n) = Pi^2/36 + Pi/(6*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = log(phi)/sqrt(5) + log(phi)^2, where phi is the golden ratio (A001622). (End)
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
KEYWORD
nonn,easy
AUTHOR
Robert Israel, Jan 02 2015
STATUS
approved