OFFSET
0,2
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). Cf. A065097 (Cat(n + 1/2), A265101 (Cat(n + 1/3)) and A265103 (Cat(n + 1/5)).
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.
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)
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 *)
PROG
(PARI) a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6); \\ Altug Alkan, Dec 07 2015
(Magma) [Binomial(8*n+6, 4*n+1)/(8*n+6): n in [0..20]]; // G. C. Greubel, Feb 16 2019
(Sage) [binomial(8*n+6, 4*n+1)/(8*n+6) for n in (0..20)] # G. C. Greubel, Feb 16 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Dec 02 2015
STATUS
approved