A337291 a(n) = 3*binomial(4*n,n)/(4*n-1).

4, 12, 60, 364, 2448, 17556, 131560, 1017900, 8069424, 65204656, 535070172, 4446927732, 37353738800, 316621743480, 2704784196240, 23263187479980, 201275443944432, 1750651680235920, 15298438066553776, 134252511729576240, 1182622941581590080

Offset: 1

Author

Lucas A. Brown, Aug 21 2020

Keywords

• nonn
• easy

Comments

a(n) is the number of lattice paths from (0,0) to (3n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/3 are the path's endpoints. - Lucas A. Brown, Aug 21 2020

Links

• R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502 (2025)

Crossrefs

Cf. A006632, A337292.

Programs

Mathematica

Array[3 Binomial[4 #, #]/(4 # - 1) &, 21] (* Michael De Vlieger, Aug 21 2020 *)

PARI

a(n) = {3*binomial(4*n,n)/(4*n-1)} \\ Andrew Howroyd, Aug 21 2020

Formula

a(n) = 4*A006632(n).

G.f.: 4*x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

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