A265101 a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5).

1, 30, 1144, 49742, 2340135, 115997970, 5967382200, 315614844558, 17055399281284, 937581428480312, 52267355178398304, 2947837630317717410, 167897169647656366330, 9643503773422181941740, 557939244828083793388560, 32486374828326106197187470

Offset: 0

Author

Peter Bala, Dec 02 2015

Keywords

• nonn
• easy

Comments

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/3).

Number of maximal faces of the rational associahedron Ass(3*n + 1, 3*n + 4). Number of lattice paths from (0, 0) to (3*n + 4, 3*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (3*n + 1)/(3*n + 4)*x. See Armstrong et al.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..555
• D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Crossrefs

Row 3 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265102 (Cat(n + 1/4)), A265103 (Cat(n + 1/5)).

Programs

Magma

[Binomial(6*n+5, 3*n+1)/(6*n+5): n in [0..15]]; // Vincenzo Librandi, Dec 09 2015

Maple

seq(1/(6*n + 5)*binomial(6*n + 5, 3*n + 1), n = 0..15);

Mathematica

Table[1/(6 n + 5) Binomial[6 n + 5, 3 n + 1], {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

PARI

a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5); \\ Altug Alkan, Dec 07 2015

Sage

[binomial(6*n+5, 3*n+1)/(6*n+5) for n in (0..15)] # G. C. Greubel, Feb 16 2019

Formula

a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5).

(n + 1)*(3*n - 1)*(3*n + 4)*a(n) = 8*(2*n + 1)*(6*n + 1)*(6*n - 1)*a(n-1) with a(0) = 1.

From Ilya Gutkovskiy, Feb 28 2017: (Start)

O.g.f.: (3F2(-1/6,1/6,1/2; -1/3,4/3; 64*x) - 1)/(2*x).

E.g.f.: 3F3(5/6,7/6,3/2; 2/3,2,7/3; 64*x).

a(n) ~ 4^(3*n+2)/(3*sqrt(3*Pi)*n^(3/2)). (End)