A337292 a(n) = 4*binomial(5*n,n)/(5*n-1).

5, 20, 130, 1020, 8855, 81900, 791120, 7887660, 80560285, 838553320, 8863227100, 94871786100, 1026317094705, 11203116342560, 123243929011680, 1364973221797900, 15207477517956825, 170321264840835900, 1916512328325665070, 21655893753689280120, 245631370596035312350

Offset

1,1

Comments

a(n) is the number of lattice paths from (0,0) to (4n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/4 are the path's endpoints.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Formula

a(n) = 5*A118971(n-1).

G.f.: 5*x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.

D-finite with recurrence 8*n*(4*n-3)*(2*n-1)*(4*n-1)*a(n) - 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-6)*a(n-1) = 0, a(0) = 1. - R. J. Mathar, Jan 26 2025

G.f.: -4*4F3(-1/5,1/5,2/5,3/5; 1/4,1/2,3/4; 3125*x/256). - R. J. Mathar, Aug 10 2025

a(n) ~ 5^(5*n-1/2) / (2^(8*n-1/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 21 2025

Mathematica

Array[4 Binomial[5 #, #]/(5 # - 1) &, 20] (* Michael De Vlieger, Aug 21 2020 *)

Prog

(PARI) a(n) = {4*binomial(5*n, n)/(5*n-1)} \\ Andrew Howroyd, Aug 21 2020

Crossrefs

Cf. A002294, A118971, A337291.

Sequence in context: A382471 A081067 A318250 * A024066 A009569 A061964

Adjacent sequences: A337289 A337290 A337291 * A337293 A337294 A337295

Keyword

nonn,easy

Author

Lucas A. Brown, Aug 21 2020

Status

approved