A265103 a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).

1, 728, 482885, 347993910, 267058714626, 214401560777712, 177957899774070416, 151516957974714281810, 131614194900668669130060, 116186564091895720987588128, 103938666796148178180041038716, 94020887900502277905668153549928, 85855382816448334044679630209920925

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 + 1/5).

Number of maximal faces of the rational associahedron Ass(5*n + 1, 5*n + 6). Number of lattice paths from (0, 0) to (5*n + 6, 5*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (5*n + 1)/(5*n + 6)*x. See Armstrong et al.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..333
• D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Crossrefs

Row 5 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265102 (Cat(n + 1/4)).

Programs

Magma

[Binomial(10*n+7, 5*n+1)/(10*n+7): n in [0..15]]; // Vincenzo Librandi, Dec 09 2015

Maple

seq(binomial(10*n + 7, 5*n + 1)/(10*n + 7), n = 0..12);

Mathematica

Table[Binomial[10n+7, 5n+1]/(10n+7), {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

PARI

a(n)=binomial(10*n + 7, 5*n + 1)/(10*n + 7) \\ Anders Hellström, Dec 07 2015

Sage

[binomial(10*n+7, 5*n+1)/(10*n+7) for n in (0..20)] # G. C. Greubel, Feb 16 2019

Formula

a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).

(n + 1)*(5*n - 2)*(5*n - 3)*(5*n + 4)*(5*n + 6)*a(n) = 32*(2*n + 1)*(10*n + 1)*(10*n - 1)*(10*n + 3)*(10*n - 3)*a(n-1) with a(0) = 1.

From Ilya Gutkovskiy, Feb 28 2017: (Start)

O.g.f.: (5F4(-3/10,-1/10,1/10,3/10,1/2; -3/5,-2/5,4/5,6/5; 1024*x) - 1)/(2*x).

E.g.f.: 5F5(7/10,9/10,11/10,13/10,3/2; 2/5,3/5,9/5,2,11/5; 1024*x).

a(n) ~ 4^(5*n+3)/(5*sqrt(5*Pi)*n^(3/2)). (End)