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

%I A265101 #37 Mar 05 2025 05:25:09
%S A265101 1,30,1144,49742,2340135,115997970,5967382200,315614844558,
%T A265101 17055399281284,937581428480312,52267355178398304,2947837630317717410,
%U A265101 167897169647656366330,9643503773422181941740,557939244828083793388560,32486374828326106197187470
%N A265101 a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5).
%C A265101 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).
%C A265101 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.
%H A265101 Seiichi Manyama, <a href="/A265101/b265101.txt">Table of n, a(n) for n = 0..555</a>
%H A265101 D. Armstrong, B. Rhoades, and N. Williams, <a href="http://arxiv.org/abs/1305.7286">Rational associahedra and noncrossing partitions</a> arxiv:1305.7286v1 [math.CO], 2013.
%F A265101 a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5).
%F A265101 (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.
%F A265101 From _Ilya Gutkovskiy_, Feb 28 2017: (Start)
%F A265101 O.g.f.: (3F2(-1/6,1/6,1/2; -1/3,4/3; 64*x) - 1)/(2*x).
%F A265101 E.g.f.: 3F3(5/6,7/6,3/2; 2/3,2,7/3; 64*x).
%F A265101 a(n) ~ 4^(3*n+2)/(3*sqrt(3*Pi)*n^(3/2)). (End)
%p A265101 seq(1/(6*n + 5)*binomial(6*n + 5, 3*n + 1), n = 0..15);
%t A265101 Table[1/(6 n + 5) Binomial[6 n + 5, 3 n + 1], {n, 0, 20}] (* _Vincenzo Librandi_, Dec 09 2015 *)
%o A265101 (PARI) a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5); \\ _Altug Alkan_, Dec 07 2015
%o A265101 (Magma) [Binomial(6*n+5, 3*n+1)/(6*n+5): n in [0..15]]; // _Vincenzo Librandi_, Dec 09 2015
%o A265101 (Sage) [binomial(6*n+5, 3*n+1)/(6*n+5) for n in (0..15)] # _G. C. Greubel_, Feb 16 2019
%Y A265101 Row 3 of A306444.
%Y A265101 Cf. A000108, A065097 (Cat(n + 1/2)), A265102 (Cat(n + 1/4)), A265103 (Cat(n + 1/5)).
%K A265101 nonn,easy
%O A265101 0,2
%A A265101 _Peter Bala_, Dec 02 2015