A265102 a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).

1, 143, 22610, 3991995, 757398510, 150946230006, 31170212479588, 6611198199648595, 1431806849011462742, 315319074704135127010, 70398290295706497441660, 15897587681946817926283230, 3624898901185998294920196300, 833406923656808938891174678092

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 + 1/4).

Number of maximal faces of the rational associahedron Ass(4*n + 1, 4*n + 5). Number of lattice paths from (0, 0) to (4*n + 5, 4*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (4*n + 1)/(4*n + 5)*x. See Armstrong et al.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..416
• D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Crossrefs

Row 4 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265103 (Cat(n + 1/5)).

Programs

Magma

[Binomial(8*n+6, 4*n+1)/(8*n+6): n in [0..20]]; // G. C. Greubel, Feb 16 2019

Maple

seq(binomial(8*n + 6, 4*n + 1)/(8*n + 6), n = 0..14);

Mathematica

Table[Binomial[8n+6, 4n+1]/(8n+6), {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

PARI

a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6); \\ Altug Alkan, Dec 07 2015

Sage

[binomial(8*n+6, 4*n+1)/(8*n+6) for n in (0..20)] # G. C. Greubel, Feb 16 2019

Formula

a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).

(n + 1)*(2*n - 1)*(4*n + 3)*(4*n + 5)*a(n) = 2*(8*n + 1)*(8*n - 1)*(8*n + 3)*(8*n + 5)*a(n-1) with a(0) = 1.

From Ilya Gutkovskiy, Feb 28 2017: (Start)

O.g.f.: (4F3(-1/8,1/8,3/8,5/8; -1/2,3/4,5/4; 256*x) - 1)/(2*x).

E.g.f.: 4F4(7/8,9/8,11/8,13/8; 1/2,7/4,2,9/4; 256*x).

a(n) ~ 4^(4*n+1)/(sqrt(Pi)*n^(3/2)). (End)